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Since an inflation system requires significant power to operate and is comparatively bulky, it is unlikely that you will ever find one that meets medical certificate requirements. My wife has one admits that it doesn’t meet medical requirements. Since we also have an automatic blood pressure cuff basefld system we set the watch to that and it seems to be comparatively close whenever we cross check it to the cuff. How they are able to get a blood pressure reading without inflation is beyond me, but it serves its purpose.This is NOT something you want to ignore (especially when it can be treated). If you’re struggling with blood pressure take a look at this: Blood Pressure Exercises VSL cbGive this a watch.The text:greetings and welcome back to smart watch ticks it’s an exciting day today we’ve got something not just new to show you but technology that is really really moving ahead we are looking at something that has ECG capability in it along with of course PPG what in the world am I talking about well you’re about to find out this is a fitness band yes and it does a lot of things like a lot of Fitness fans do but it does them better leave supposedly it does them with pizzazz P I is easy right the zebb oh come on give me in here I’m wasting everybody’s time we’re not gonna the ECG heartrate smart band called the T six and this one comes to us directly from the manufacturer themselves totally duh and we’ve got a scoop on the inside workings of this stink what is it and how does it really operate let’s take a look at the specs it’s a ECG bracelet using ECG PPG medical grade components it does blood pressure heart rate monitoring it’s totally waterproof sports snorts what what health bracelet Model T six running Bluetooth 4.0 covering those different languages the app we’re gonna link to is brand new go hard I haven’t seen that one before it’s support for all of these good things okay as an exercise plan call or message it’ll show those it’s the screen size the battery capacity standby time seven days two hours to charge and basic material is PC all right more on that in a minute but first let’s get into it let’s take a look at this thing when we open it up we get the manual in Chinese hopefully English as well we’ve got a little charging thing which is the standard we’re seeing this a lot on fitness bands it’s like a little clothespin that clamps onto the device with some charging ports and here it is this is the unit got a little plastic cover that we can take off now anything else no that’s it let’s look at this well there’s one big glaring silver button in the front and most of these devices have a button to navigate with oftentimes it’s invisible but this cop this time it’s metal and why is it metal because on the back we have two metal plates and why do we have these three metal plates so we can make an electrical circuit between your left arm around your wrist and your right arm on the button and electricity will loop between your hands through your arms and across your heart and that’s what’s important because this picks up the electrical signal from your heart this the diodes right here for your pulse rate that we’re used to this picks up the movement of blood in your capillaries and it can calculate your pulse rate from the change in that blood movement but this this is what makes this thing really cool this is ECG let’s take a look at the manual in English it says all of this thank you for purchasing it okay the bracelet elements goes over what those electrodes are application download this is where you get the go heart app for i. OS or Android but you can just go to the Google Play Store right now and download the app you’ll need to create an account and log in but once you you do that you get access to all of the stuff and you could check it out even before you order this thing it’s got sports mode health mode there’s information about it step one and step two how you actually use it and what’s really fun is you can get the ECG data playback and save as a PDF file your own heart rate just like at your doctor’s office and then there’s the care how you care for your band and that’s it on the English side on the other side you have the manual all in Chinese and zip so let’s turn it on let’s check it out let’s see what it can do I don’t know I don’t know what to say I’m I am tongue tied by this one and that usually doesn’t happen it really doesn’t but this is incredible and it’s even more incredible when you see the app and you’re about to let’s walk through it real quick time haha 24 hours it’s 3: 50 5: 00 in the afternoon 12 hour mode it’s available your steps not many but just to start your calories burned computed by all of the data you put in the distance you’ve traveled not by GPS but by calibration and calculation of course from your information then you get into the ECG plus BPG whoa it’s flashing down below here’s the PPG and here’s your blood pressure and here’s the temperature in degrees centigrate haven’t figured out how to change that though here’s what this data looks like i’m gonna press and hold for a moment it flashes and now it’s accumulating some baseline data before i let go so I get a really good average to start the reading with there you go now it’s calculating the heart rate or pulse using just a green diode and it vibrated and there’s my reading 73 come over here do the same thing I’m gonna press and hold get a little flash getting calibrated let it go should take off on its own in a moment and now again using the the diode technology is calculating my blood pressure notice I’m not touching the metal plate so it’s not in involving the ECG this is just PPG 130 over 83 when you go in here and you actually touch the button where it says ECG plus PPG it’s gonna go for about 30 seconds and it’s collecting the data the same as you’re gonna see in the app in just a minute and after the 30 seconds instead of displaying the results on the screen because there’s a lot of results it’s gonna basically tell you I’ve got it it’s been recorded and next time you sync up it’ll send the data to the app and then it cycles through again so all the way through oh and messages if you have any and find your phone if you’re tethered and then the overall information and a power off button it looks like I’m going quickly because it times out and then it loops again if you press and hold you change to the other clockface which is an analogue one that’s it and you go through them all again and again press and hold and you’re back to digital so you can see all of the data for instantaneous results on your smart band however when you tether it to the app oh my gosh look at this app Oh it’s called go heart hey anything that tethers to go heart buy it in a heartbeat haha pun intended this is the first one I installed it and I did everything and then I’ve uninstalled it so I want to show it to you right from the very Get. Go we launch go heart out and you can create an account or you can log in with any of these third party apps Google’s not there which is sad because I usually log in with Google but if you have any of these account you can login whoops and you can also login once you’ve created an account which is exactly what I’m going to do right now now when you log in successfully which we have done you come to this first page which is your overall summary of sports sleeping and PPG heart rate remember PPG is your optical heart rate or pulse we’re gonna call it pulse rate measurement a con yet heart rate here you hit the health tab and you get to this very busy looking page you hit this one and you get to care which is where it’s loading any friends you have that are logged in on the network so you can I guess work with and encourage each other and you hit mine and this is where it has your basic data and I previously when I logged in set up Smart Watch ticks as my username and I took all the default settings for the case of this example and important it lets you wear it on the left hand or the right hand and believe it or not the ECG chart that you’ll be seeing shortly will run forward or backward depending on which arm you’re actually wearing it on Wow okay security settings you can change your password or you can log out again so you’ve got all of that basic stuff what’s the ability to set an icon if you want to and connect a bracelet now the bracelet is on obviously you saw it showing its display here a little bit ago it’s four o’clock in the afternoon I’m gonna say connect a bracelet bluetooth is on and look how fast it finds it and look how fast it connects it nobody else has done that like that it is on and it is connected and we’re ready to go the time would be synchronized automatically if it hadn’t been and we can go through all the different settings of things that we can do let’s do that user settings we check that out that’s all of that stuff now target heart rate our target steps is where you set whatever step count you would like to accomplished in a day and it will monitor all that through the day and reset it midnight your sleep target is how many hours of sleep you want hours and minutes and you can confirm that your device management is where we’re going to get into shake snapshot actually turns on a camera and it has the option for you to tag your photos with location if you want to and I’m going to just wiggle this a little bit while I shake the band and up to one and boom is taking a picture so skipping you a countdown to so you can set it up to do selfies and watch it while it’s happening so it’s a different kind of its own camera in here it’s not linking to an external camera app is working with its own camera app within the app itself fine bracelet is gonna cause it to vibrate and it is I’m gonna set it on the camera here I don’t know if that’s making sound or not but it’s lighting up an icon and vibrating the bracelet and then there’s operating instructions which we’ll come back to so device management here’s where it gets fun for normal use of notification services you have to open the go heart read notifications permissions and that’s right here and this tells you all the stuff that is gonna happen and this is standard Android information as you know you have to enable these things in order to get push notifications so we’ve got go heart set up I can turn on an hourly heart rate monitoring process I can and now this is interesting set lift the wrist to brighten the screen or I can set turn the wrist to brighten the screen they’re different and that’s the first time we’ve seen that I can have a call reminder so um it will push when my phone rings to notify me on the band that a call has come in or a text message and I can set do not disturb looks like it’s locked into this time frame twenty two hundred to eight hundred I don’t see a way of being able to change that but then you won’t get any vibration while you’re asleep and you’re in your band if you’re sleeping with it to collect your sleep time and more reminders the course will take you into all the different app apps that they’re supported and give you a reminder from those plus all of these sedentary reminder type things like taking medicine meetings your overall sedentary remind you can set the interval the start time and the end time for any of these so you have multiple recurring alarms that’ll vibrate on your wrist when these events come up and they’re selectable by you four of them which is which is nice too we got a lot more robust capabilities there’s individual alarm clock settings that you can do by adding an alarm at what time you want it what days of the week you want it and confirm that and you’ll get vibration when the alarm goes off now here’s the fun 12 or 24 hour format we are on 24 hour format right now I can set it to 12 hour and I can set my measurement to imperial instead of metric and if I do this hopefully I’m at four do you see it change from 16 to 4 my brightness is a little a little off that’s why I oughta matic brightness on the cameras setting it to be a little hard to see there but it’s in 12 hour instantly pushed from the app to the watch you have firmware that you’re running and it you cannot presume update it if there is any update to come and then you could of course disconnect the device so lots of robust features in the device management and that’s it except for operating instructions in the last tab let’s run through those operating instructions because this is where we’re gonna start getting into the important information about what you can do with this watch our frequently asked questions talk to about health data measurement first of all you press on the metal electrode of the device surface when you click the app one key measure for accuracy please be quiet for five minutes and no smoking drinking coffee food or drugs within two hours before your test now this is if you really want to get a good solid baseline and you’re gonna want to get about that baseline in order to do the calibration you’re gonna see that to be still stable and quiet during the test and please try handheld or wet the electrodes if the ECG is disordered or has no wave okay keep electrodes closely touched with the body that’s about the health data measurement change phone to operate when your Android phone connects with a bracelet if you use another phone this is how you deal with that if you’re switching between i. OS and Android you have to kind of do a little bit more and restore the factory settings before going back to Android different operational tactics so if you do both Android and i. OS read about that one now why need to measure high blood pressure after input blood pressure reference value calibration there is relation between body blood pressure and calibration value we can check body blood pressure through throughout throughout calibration value high blood pressure and low blood pressure have obvious differences due to individual physical differences so pressure patients can get more accurate data if measured after setting the calibration value I think what they’re saying is if you’re prone to low blood pressure and you calibrate it for that then the measurement it’s going to take is going to be more accurate because it will deal with that differently probably in the way it assesses the wave is it is it calculates all of that then it would be if you are prone to high blood pressure so you can do it just taking the normal values but if you calibrate it first you’re gonna be more likely to have your accuracy and calibration is really cool on this thing it really is okay reasons you might have inaccurate data is the wearing man you want to completely touch with your skin and keep your hand and heart at the same level this is a blood pressure monitoring thing so you got to do all those kind of things if your hand has many hairs well your wrists really please add water to make it wet and be quiet when you do the measurement okay all right ECG introduction e electrocardiogram in each heart cardiac cycle the pacemaker atrium and ventricle have been excited accompanied by changes of bioelectric through ECG recording from the surface leads to a variety forms of electrical potential changes which is short for ECG ECG is objective indicator of heart exciting spread recovery process there is obvious difference between action point bitmap recording from single excited cardiac cell and each heart cardiac cycle ECG got that good just basically you’re Rhett measuring your electrocardiograph information from collecting data through these three metal sensors on your skin and this is where I was saying depending on if you wear it on your left or right arm the silly chart might actually go backwards so if you’re left handed make sure you have said in the settings that you’re wearing it on your left arm so I’m gonna go back over here to the app I’m gonna come over here to health and I’m gonna pull down and let it synchronize now it’s sending over not just the numbers of the readings that it got but it’s sending over that entire chart which you’re gonna see here in a minute I oh if I go over here to sport shows the steps I’ve taken I haven’t done the sweeping yet and there’s the PPG heart rate information average sleep heart rate highest lowest all this kind of data can accumulate over time back here I wanted to give us time for it to come in so it’s measuring a hundred and seventy eight beats per minute is getting a hundred and twenty four over eighty as my blood pressure and here is my heart rate analysis when I come in here it’s saying that I have no abnormalities in terms of the heart rate variability heart rate variability is the distance between the heart the heart beeps right if it’s not eccentric then I guess everything is fine and it anyway it’s basically telling me my ECG has no abnormalities but that’s not very much information let’s take a look at it here’s the actual reading of my heart rate as measured by the band when you saw me collecting this data and it shows it to you live reading here now not live but recorded it’s actually been recorded it’s not showing me on the band directly the live of course because it has a too small of a display but it’s showing it to me here across the screen but wait there’s more health report generates the whole thing there it is mr. smart watch ticks Mayo age 24 height 170 wait 65 again this is basic information that’s adjustable any way you want to match you the average heart rate through the whole 30 seconds with 78 beats per minute here’s the tag information across the bottom and boom I can hit that button in the upper right hand corner I can email it out I can post it to Twitter or Facebook I can do all sorts of things with it because it’s available to me and I can actually share it as a PDF so when you share it for PS PDF it can be opened up by your health professional if you email it to them in a standard Adobe Acrobat kind of a way you know any PDF reader would be able to display the intricacies of your heart right now I can’t pinch and zoom I can’t like expand it in this format on this phone but obviously if it’s a PDF you can expand it all you want to and your health provider knows about all these little subtle things in here beyond what you’re looking at here cool huh well that’s all fine and dandy but it’s not calibrated no I didn’t actually calibrate this one so I’m gonna go back here and you see I can’t take a measurement from the app unless I calibrate first so I’m gonna do that as excited and hyper and wound up as I am I’m gonna pretend right now that I am relaxed just got up I’m gonna set my baseline heart rate by tapping calibration and here’s where you got two different options you can go to a graded calibration where you say I’m medium or medium rare or rare medium well or well done you can set your grade of calibration if you think you’re typically a little bit high you know even though the doctor says you got to be on meds or whatever you could choose that and you could confirm it great we’re not gonna go there we’re gonna go precise I mean three sighs okay enter a medical value 120 over 80 I think is typically what you get back space I don’t know AED confirm that and then it says for accuracy please be quiet for five minutes and no smoking drinking we all saw that a little while ago right keep the electrodes closely touch with your body confirm are we ready okay I’m gonna stay start measuring look I’m in another screen now where it’s actually showing these let’s move this over a bit the Lighting’s gonna be kind of funky here because I need to keep this right but I need to also be able to show you the band how can we get this all in the screen together because I have to actually have the band on okay I’m gonna say start measuring says poor contact I’m gonna touch it now it’s got contact and there it goes it’s actually showing you in the top the ECG from the electrodes and in the bottom the wave that’s being created from the optical reading the PPG now later at the end of this video when you go through the technical details of how these two get married together to give you accurate data you’re gonna learn about a p WT T pulse wave Time Tracker or something or other anyway and that’s really really cool there I’m calibrated I’m fully calibrated the whole heart wave has been calibrated now so that if I do a measurement from here not from the band now and I’m actually measuring it from the app and confirm it start measurement I got four seconds to get my finger over there it’s there and boom now it’s actually doing a measurement and that measurement is highly accurate because it’s calibrated to go inside coincide with my default level that it just acquired when I created my default measurement and that’s really cool I mean this is they talk medical grade and I’m still gonna say entertainment purposes only folks don’t rely on this for medical advice but Wow check it with your doctor check it with your health professional make sure that everything looks good you’re getting similar readings and everything and you’re getting a whole chart you’re getting your whole ECG PPG chart information here that’s just been measured it’s all in here in my data now if I go back to my sports I’m not sure if it has to wait for the end of the day I guess it does to calibrate all of this yeah I can’t update it but you can go you know back and forth by day you can get your your data all in through here when you’ve had enough data but here in the health thing okay it’s showing my showing that my heart rate variability health index right now is good it’s at 80 it’s not like really low which it would be if I was quiet and mellow and it’s not like way up here other way around it’s not optimal it’s not general I guess higher is better the optimal but it’s it’s giving me my health index okay and here’s my heart rate information over the past couple of measurements here’s the heart rate chart information it says 78 is where I’m coming in heart rate under quiet condition yeah and then the other the other one you’ve got the ECG blood pressure 119 was I guess the newest measurement the first one that I did earlier before calibrating was right here let’s take a look at this one here’s the systolic showing me 119 right in that range over 76 there’s those charts great great deep they are coming through on this app and this one again is where you get the actual charge themselves so it’s by newest at the top it loads the charts and you can start playback of both of the graphs right there at the same time you know it doesn’t uh it doesn’t turn and go the other direction unfortunately it’s just gonna show it to you this way but again it’s not a problem if you just want to take a generic look to make sure that the the data looks correct that’s great but if you want to see the whole thing you go to the health report let it generate it and Wow there it is there it is create a PDF email it out to yourself archive it send it to your doctor whatever you want to do with it all from this tiny little band all from this thing whoa nice huh now I want to you one last thing before we wrap this thing up I’m gonna come back here I’m gonna go and do another measurement and get ready here we go I want you to pay attention to this thing because I start the measurement poor contact now I’ve got contact I got my hand on the button okay it’s doing it right now can you see me over here can you see the chart I’m gonna wiggle it a little bit I’m gonna mess around with it I’m gonna take my finger off poor contact good contact or contact good contact wiggle wiggle wiggle wiggle we now you see the thing that’s on the screen of the band is certainly not moving around that’s just an icon but what it’s collecting actually here okay it’s giving me that data which is erroneous I am getting erroneous data here it’s saying heart rate is too fast but the date is all screwed up so I’m going to confirm it so that you can see it and I don’t want to start a measurement here I want to go back to the data itself heart rate too fast see that and I’m gonna come in here and go to my health report because I want you to see what the data look like you see the times that I was messing around with it this thing is real folks it’s real yeah all right Wow do I have a source for you on this if I do I’ll have it in the show notes this is coming straight from the manufacturer are the producer the company that’s created it told Leda so your feedback in the comments section about this device will be going and I’m sure read directly from the company they’re not in the business of selling these things they’re in the business of making them so whatever you like or don’t like this is a place that you may actually get the ear of somebody that matters all right so check the show notes for information we’ll have buying links for this if I can get them for you I think they’re gonna be distributed through our normal channels soon yay now we do have a supporting partner banggood check the show notes will have a buying link for you down there with a discount price on the t6 so how about some real life data I’ve spent about four days with a band and I’ve got some really good data to show you and the data you’re about to see would be similar to if not identical from any of the supporting bands or watches that are professing to do ECG plus PPG measurement technology they’ll all be working with apps from this particular company now let me show you check your app when you’re looking to see what it tethers for the band that you’re interested in buying make sure it’s either where hard or go hard or one or more of these other ones okay that’s because these are the type of apps that will let you capture your heart wave and they look somewhat like this there’s a page that shows you information another page where you do your calibration and measurement of all the ECG information something about connecting with friends and then your overall settings looks a little familiar to what we just saw right this is the where heart app now if I take us into the go heart which is the one that I’ve actually been using and you’re about to see the data on you’ll notice that it has the same concept layout but with pretty pictures it’s been dressed up a little bit but it’s pretty much the same app so you could use where heart or go heart and I believe there’s at least one more in here let’s try fit live and look you’re getting the same display so we have a variety of different apps by different names that do exactly the same overall function so check it out like I said we’re gonna actually use the go heart app which is where I have all the data on the band that I’ve used for collecting it roll to refresh and there’s my data so far for today let’s dive into this the sports shows you hour by hour against the bar graph with no measurements on the side how many steps you’ve taken your calories burned in your mileage if we look at yesterday here was my distribution from midnight to midnight and how many steps and I go back another day and I’ve got that so that’s some of the information that we get if I show you on a weekly basis you see the total weekly steps broken down by day and total mileage and calories burned for the week or for the whole month interesting huh okay when we look at sleep this is the sleep monitoring of the night before so this is last night sleep six hours 37 minutes deep light and how long I stayed up which I think was when I got up this morning decided to lay back down and hold real still and pretend to be asleep Shh don’t tell anybody but I got some extra light sleep by pretending holding my arm real still but the awake was you know getting up and moving around getting some water going to the bathroom and then coming back on link down for a little bit or about 702 but when I got up after that I was up and it logged it as six and a half hours and my sleep target of 82% wanting eight hours and wake up time three I have it on different days he was the night before the night before that I didn’t wear it on purpose and I don’t have any erroneous time this is a good check when you have a device to see whether or not you get bogus information when you’re not even wearing it and the night before is when I actually started wearing it so I got three days out of four what’s it looked like weekly sure enough I got the bar graph showing the hours and my average weekly sleep time and the time I stayed up intermittently between the time of going to sleep and waking up and then of course there’s a monthly again my monthly average time really slick huh okay that’s a sleep time then we get into here and this is the what did they call it PPG HR remember this is optical and this is heart rate or pulse so this is the last pulse rate display of the every hour that I’m accumulating my heart rate right now and these are the this is the information of the heart rate that it was able to detect from low to high an average and my last night’s sleep average heart rate yes it was doing that while I was asleep yesterday I took it off for a while I did some really good exercise I guess well this is highest was 95 I have a day that I did more exercise so let’s look at that here we go 102 well that was at midnight maybe it was like fumbling around sometimes you’ll see an erroneous one pop up you just got to kind of ignore it the day before that it’s here and then I think that’s about it no more data look at that it doesn’t just go back and show you zero it it knows that I didn’t start until then alright I guess that day with 102 was my workout day so that’s this first screen your steps you’re sleeping and your heart rate then you get to the brand new stuff we’ve seen this briefly before and what you just watched this is the accumulated data now I have the calibration we went through and then I can do a measurement directly here or I can do it on the band capture it there by clicking on the button that says ECG plus PPG and remember it’s said to the app well believe it you have to send it to the app you have to sync it up after you take that data because if you go and do another measurement it wipes the other one out and puts in the new measurement in the watch you have to transfer them one by one I found that out don’t like think you can do three or four of them during your run on your watch by itself and all of them will transfer to the phone it won’t just the most recent one so if you’re gonna do the readings of the ECG Plus PPT on the watch or on the band make sure you transfer it to your phone before you do another one this one doesn’t matter because you’re doing it right on the device itself you just tap measurement gives you the countdown and you can collect the data right here and what are we collecting we’re collecting the ECG heart rate now this is different than the PPG one and when you look in here I haven’t done it at all today but if I go back to yesterday these are my readings 80 65 64 when you go into it it talks a little bit about where you are in the heart rate under quiet conditions alright doesn’t change so it’s just kind of giving you a information sheet and your total up at the top not a lot of data there so you could just work off of this if you wanted to the ECG blood pressure nothing today but there’s the ones from yesterday and again if I touch it it tells me a little bit about systolic low ideal high and diastolic and the chart under low pressure and it gives me my readings systolic and diastolic and again if I took another one this hasn’t changed so this is information but there’s your data which again you got right on the chart itself however the third button this is the one that your entire ECG way cool this was the one and I don’t have any data again for today but here’s the three that I had that we just looked at from yesterday and if I go into it and I showed you this already I can start the playback of that particular one and over time looping back on itself you’re seeing simultaneously the ECG reading and the PPG reading right and it’ll do that for the 30 seconds that it’s accumulating or you can hit the health report it generates it and you get the chart now I saw something that was like really interesting when we were doing the setup and it said ask you if you’re wearing it on your left or your right on because it made a difference it said that these little Peaks that you see going up will be backwards or upside down if you don’t have if you don’t wear it on the right arm Wow mr. ticks is of mr. ticks if he doesn’t check that out right so my first two readings of this day I had the band that’s on my left arm actually on my right arm and I did the reading anyway and sure enough here’s the report and look at it they’re all upside down how the heck did it know that I have no idea but it’s getting the electrical signal you know from touching and and the plates on the bottom and their top and the finger and all of that anyway the electricity going through goes through backwards upside down when it’s on the wrong arm now if I would have said I’m wearing it on the right arm it would have flipped the chart and everything would have been fine so that’s an interesting thing to note now also very curious about how does it work as most of these have been sedentary how does it work under stress conditions so I’d go back over here and you see the heart rate too fast that’s when I was working out so I get different charts now when moving now to be fair I didn’t continue running and stuff and tried to do it while I was running I stopped I held real still I was breathing heavy but I took the thirty Seconds to accumulate the data and this is what I get looks like a normal heartbeat right I mean I’m not really looking for it to do anything weird there’s a little movement glitch probably a couple of places but all in all the chart looks good but I imagine a healthcare professional and I really wish I could pinch and zoom but you know I could just do a freeze frame of this and pinch and zoom it right let’s see if I can do that press my buttons I love the LG g3 because all the buttons are right on the back and I just press the top and bottom and it makes my screenshot let me bring that up for you here we go now you’re just looking at the screenshot that I have of that page but now we can’t really get in here and this is what I’m talking about folks I am NOT a trained medical professional but those of you who are take a look at this is that giving you information that could be used in a way to help your client I hope so at least it’s a movement in the right direction to where we’re now able to calculate the heart rate by the time delay between these Peaks which appears to be much more accurate than the thump thump of the blood moving through the capillaries this is electrical it’s really fast and it’s right there plus it’s got all sorts of other information you get all your basic stuff here and again you can make this a PDF and share it send it to your doctor whomever this is my heart under light stress when I was getting a pulse rate somewhere on here of a hundred and five ok now we’re back on the app so if I go back again it’s a my heart rate was too fast as their assessment and I take the second one just to show you four compare since there’s another one that I did okay get to feel for that and I’m gonna show you a nice calm one just in case you’re interested in seeing if there’s a difference late at night just before bed there you go actually it looked more regular to me when I was exercising I wonder if that says something I think I’ll exercise more interesting I gotta admit this band and this technology has gotten gotten me up off of my perch him out doing some great movement and I’m gonna stick with it so that’s that’s a benefit in and of itself for a little forty dollar band amazing alright so what else have we got in here every day is captured however many times you do it it’s not automatic you have to do it manually you can do it from the app or you can do it from the band and make sure you transfer it now it’s going back in two days that I don’t have any data whatsoever so in this case it will just keep going it doesn’t know if I have when I got the band actually but when I actually do two measurements they’ll show up the heart rate variability index seems to be just the latest reading I’m not getting in HRV in the analysis of any of these are you seeing it anywhere I don’t think so on the health report itself it’s got all of that which is your basic stuff and the average heart rate the strip chart information and the date and time so the heart rate variability is an area that I think personally they could really improve upon because there’s a lot of meaningful information there as I understand it heart rate variability is the change in distance between each of those heart beats if there’s a lot of variation than you’re under stress and if it’s very uniform then you’re calm and there’s a whole bunch of stuff related to heart math and other things that go into heart coherence and I mean it’s a big deal and it’s really great for those dealing with wellness so this is the first introduction of ECG in a wearable concept get ready to see much more expansion in heart rate variability and that pretty much covers it the care again I don’t have friends so I don’t have anybody in here but as I understand it your data can be cross shared with other people who either have the band or a similar one like you have or maybe even just your relative you want grandma and teach her how to do the measurements that’s great perhaps this is where you can see her results by setting the app up and logging in and marrying it together and grandma will show up as a friend and then the last tab we’ve seen before is the overall settings of how you go through to work with the band including setting your sleep target and your steps that’s an overview of the data supported for these ECG plus BB G devices so what is all this ECG and PPG and P wtt stuff anyway and why do we care well we’re gonna explain that in this little section and if you’re watching it it means you just watched a video about an ECG PPG sensor fitness band or watch because I’m gonna attack this section at the end of every single one of those since its new technology and I know you guys aren’t watching all of them if you’ve seen this segment before you can bail out of the video because it’s a repeat if you haven’t seen it you might want to listen to understand about the band or watch you’re considering purchasing I’m gonna read through it with you first of all take a look at these pictures you know in the medical community they can hook you up to a biomedical device that creates this whole heart wave chart labeled with ABCDE in all these different areas and Heights and things that the medical folks know how to read it’s a medical a equipment device and is relatively expensive although you can probably rent them for the time you need them if you have issues related to your heart you have electrodes that go on your body and they pick up a signal that gets recorded through the electronics in the device well the electro cardio electric monitor is a bio electricity that’s generated from each of the heart sections beating by a pacemaker atrium and vintage Giller yes the ECG data is calculated from the changes of it you’re gonna find that this is kind of Chinese English as we read it as well so I hope we can both figure out what they’re actually saying because the sentence structure isn’t a isn’t perfect what they’re looking at is is taking this concept and shoving it into this device on your wrist with three electrodes who are usually on the bottom and the third one is something that you touch on the side without letting your left and right arms make contact okay so that the signal that’s being picked up by those electrodes will come from or pass through your heart from the bottom electrodes and separately from the top electrodes down both arm according to the principle of medical testings you put a finger on the electrode plate of in this case it’s the t6 one of the watch bands were reviewing but any of them that are designed this way while we’re on risk to produce bioelectricity through the body circulation to detect the data generated the goal is to get a relative chart similar to this one that comes from this now we’re going to find out why you need that first we have to do some definitions the definition of ECG is electrocardiography and there’s ECG sensors that measure the bio potential generated by electric signals that control the expansion and contraction of the heart chambers PPG is what we’re used to seeing for getting your heart rate on smart watches and smart bands PPG here we go is photo lenses plan fuzzy plan fee Smaug Rafi length the Smaug Rafi photo Flint a spark Griffey sensors don’t do that with a mouthful of jellybeans these use a light based technology to sense the rate of blood flow as controlled by the heart’s pumping action those are the little green lights you see on the bottom of your watch and that’s what we’ve typically used to come up with your heart rate your pulse and I guess in the sort of newer technology it’s been used to create your analysis of your blood pressure and your blood oxygen but we’ll continue on that one because we still haven’t been able to fully test the accuracy of the PPG by itself what’s really cool and what’s new with these devices is to blend the combination of the ECG and the PPG for a PW TT that lets you come up with a better accuracy of all of your biometrics okay comparative questions let’s run through these that tells you the difference between the two so you can better understand them how is heart activity measure ECGs sensors directly use electric signals produced by the heart activity while PPG sensors use electrical signals derived from light reflected do the to the changes in blood flow during heart activity I kind of got that from the definitions right well which is most accurate ECG is a reference standard signal that is used for monitoring cardio health and wellness by healthcare providers PPG sensors on the other and typically use ECG signals as a reference for static heart rate comparison kinda sayin that ECG is sort of the standard that is used by healthcare providers and the optical method is measured against the static for comparison to see how accurate they are hint ECG sounds like it’s more accurate Kempe heart rate HR be measured heart rate is the same as pulse width ECG HR can be measured accurately HR can be measured with PPG however it is only suitable for average or moving average measurements that’s why it’s kind of difficult to tell when you’re getting your heart rate on your current watch if it’s real or not it’s an average so if you’re sprinting and checking your heart rate at the end of the Sprint it may be reading lower than it actually is because it’s doing an average or what’s called a moving average which means throwing away the oldest data and say taking the last five or ten measurements and averaging all of those as it moves forward in time that’s the moving average okay can’t heart rate variability oh my gosh here’s another term what in the world is that can it be measured HRV can be readily derived from ECG data as our peak intervals can be extracted wake up HRV come on and there can be read heart rate variability is the how much variance there is between heart rates let’s say you’re at 60 beats per minute well what if you got 60 63 58 60 that’s the heart rate variability correct me if I’m wrong any medical providers I’m not a doctor I don’t pretend to really know this stuff I’m just working my way through it like the rest of us here heart rate variability can you measure that and apparently a measure of the heart rate variability gives you an indicator of your level of stress if you’re calm if your xeni if you’re doing mindfulness it should be regular okay your HRV should be regular if it’s static and changing it could mean you’re under stress so that’s the watches and devices to tell you your stress level that’s probably what they’re attempting to give you is the HRV which apparently is really hard from the PPG because if that’s a moving or static average right and doesn’t really take into account the variability we move on HRV can be readily derived however from the ECG data as our peak intervals can be extracted with millisecond accuracy our peak a think is that little bit you know the little sound you see in the TV shows went before you go flatline right the little peep peep that’s the our or what you call it our peak and you can measure that with millisecond accuracy which means you can really tell the time difference between those Peaks right so that meaningful HRV data can be obtained with short duration measurements usually like 30 seconds or so which is typically what you get in the chart that comes from the transmission of this data from your device to your app in your watch I mean in your phone that you could extract and send out to your healthcare provider with PPG sensors peak interval accuracy is limited by usable sampling rate due to the high power consumption of the LEDs with PPG pulse rate variation correlates with h RV alright with PPG this is the light diode kind of thing P our V pulse rate variation correlates with the heart rate variation for longer periods of measurement over five minutes but not for short duration measurements because it’s doing a moving average only if the moving average starts to change dramatically which you can only get over a long period of time in a sense you’re looking at the average of the averages as a standard deviation from the average of the averages will you find that you have stress or not by trying to use the PPG diode technology to get pulse rate variation that’s too long to go by goodness by then if you sat for five minutes you were to calm yourself down you wouldn’t need this maybe that’s what they’re trying to get you to do breathe how long does it take to take a reading well you see G’s sensors don’t require long settling times I mean you hook it up and it goes right stat clear so meaningful readings can be obtained very shortly after startup PPG sensors require a relatively long setting time due to the need for you ready measuring the amount of ambient light and calculating the compensation needed for canceling its effect are you taking your pulse in the dark in the sunlight are you moving are you moving in and out of shadow and all that ambient light has to be added or subtracted correctly to the tiny variation that the light diodes are getting back from in the sensors that represents the blood movement through your capillaries it’s really tricky and they got big old algorithms to try to cancel out the effects of ambient light but nonetheless it can’t really start giving you a good measurement which is why you don’t see it right away on your watch until it’s measured the ambient light and compensated for it whereas the ECG boom right there PPG sensors may also require compensation of motion artifacts to produce a reading come on common sense if you want the thing to work you’re going to have to hold really really still when you have the optical sensor reading your pulse if you move a little bit your capillaries are moving underneath and it’s just gonna throw everything off you found that out and that’s why often times we think these things are not accurate well they’re partially accurate as we’re learning and they’re not accurate because you’re not standing still and your ambient light is changing fix those conditions and your current watch will be or accurate there you learn something making it worthwhile watching this whole silly thing right all right wait there’s more here we go which has the lowest power consumption we need to know that for these darn low battery watches and bands right neuro skies beat eem 101 ECG sensor features extremely low power consumption operating it two and a half millions most PPG sensors require approximately thirty milliwatts of operating power well as you know that’s really not all that much either of those in comparison to cellular radio or GPS so you’re not like really draining your battery unless you have it on continuously monitoring your heart rate probably with either sensors but for sure with the optical which is the smallest footprint well a BDM 101 ECG you think this is a sail sheet for the BDM 101 probably is it it’s the world’s smallest ECG sensor at only three millimeter by three millimeter with no additional components required PPG sensors featured a relatively large PCB footprint that’s not the PCBs that are toxic I have no idea what it is but I don’t hopefully it’s not oh my gosh if we’re warning around with PCBs on our wrists it’s a big package typically six by six which is like 9 times bigger a turd four times bigger plus additional PCB area for the microcontroller unit battery management crystal LED circuits and don’t forget the little wire that you know has to hook it all up a lot of stuff going in there it’s a big footprint in a small package it takes up more space but they’re not mentioning though is remember you got to have electrodes and so it doesn’t take up a lot of space but there’s two plates on the bottom and one on the side or the front that have to be touched so that’s all part of the ECG circuit but the little goody itself that figures it all out is a tiny little thing on the ECG is any external equipment required to produce a reading well on this fancy B MD 101 sensor features on chip it features on ship HR measurements so no external equipment is required in contrast the PPG sensor does require external micro controller unit for HR calculation this is all in one this one has a little separate diode thing and that’s why you got that that cable inside if you take your watch back off that hooks it in to the main circuit board because it’s got to hook up to a chip to read and understand what that optical sensor is picking up how easy it is to integrate with a device the medium 101 ECG sensor features an SDK Android i. OS for easy app development and integration which is folks why we’re now able to get this stuff on these smart devices so far it’s in bands if you know if you’re watching this video you just probably have saw a review on a fitness band that has ECG capability but it’s gonna come to watch as you mark my words it’s by 2018 we’re gonna start seeing ECG ppg integration on watches as well why not it’s small it’s just a matter of isolating the case and putting electrodes on the back and someplace to touch on the on the watch to get the electrical hook up the sensor features an an SDK which is sort of like an apk after you put the SDK in the apk you got an app that you can use right for easy app development and integration PPG sensors require mm an MCU firmware up development and integration in addition to host platform app development and integration which is why we’ve seen a lot of the different watches more or less accurate or not around the optical sensor stuff because the sensor unit itself is just spitting out the data it’s all up to the MCU firmware that interprets all of that stuff to give you accurate or not so accurate heart rate or pulse data okay I guess we should be saying pulsed when you’re talking about optical and heart rate data when you’re talking about electrical get the terminology right when it comes to key considerations like size power consumption accuracy ease of integration and richness of data ECG biosensors present a clear advantage over their PPG counterparts for heart rate monitoring but it’s just now hitting the market in our tiny little watches and bands ECG biosensors are capable of delivering the widest range of heart health metrics because they are able to capture more comprehensive signals of overall heart performance including both heart rate and heart rate variability now up until now all of this we are talking about your pulse rate your heart rate nothing yet about blood pressure or blood oxygen two other measurements that the medical community uses and fitness zealots used to understand their body dynamics let’s move on that brings us to pulse wave transit time PW T T and its relationship with systolic blood pressure how do you take the stuff that we’re getting from the heart wave and the biometric optical sensor alright the ECG oh sorry I’m off the screen ECG signal and the PPG signal how do you mess around with that in order to come up with blood pressure in relationship to your actual systolic blood pressure to get accurate readings well this paper talks about all of that and I’m gonna put it on the screen and let you read it maybe we’ll jump into a little bit of it but and I also have the reference to it in the show notes down below and hopefully you can just freeze your screen right now and read this I’ll give you some overall let’s look at the abstract to get the premise of where it’s coming from it says the blood pressure is one important parameter in assessing a patient’s or your own health status and fitness and so forth the measurement of blood pressure using conventional method is less practical in is is less practical in monitoring continuous blood pressures this paper proposes an automatic technique for monitoring blood pressure continuously using the relation of the pulse wave transit time with the systolic blood pressure the measurement of PW T T involves the marking of electrocardiograph R the ECG our peak from electrocardiograph signal and the peak of peripheral pulse wave from the p PG signal okay that’s the optical and the electrical data collection involves three different sessions which are resting period exercise period and recovery period thus establishing the relationship between s b p and p WT t the results of monitoring blood pressure based on pulse wave transit time will facilitate personal healthcare monitoring blood pressure continuously and that’s why i do deep breathing exercises ok introduction gives you some of the background basically what the p WT t really is and then we come back up here to column two and it continues with that chart and you can kinda see that what they’re doing is measuring the time this is against time the time between the peak that is seen on the ECG and the bump that you see on the optical PP g this is your systolic here my wristers love it the this is the optical pump bump that that is used to compute your pulse rate with the optical sensor and this is the really fine detailed microsecond accuracy of the ECG getting both those signals coming in out of your body at once and analyzed can let them compute your p WT t and this whole paper is based on you that PW TT with your systolic blood pressure to get biometric data that’s more accurate for you and your body so it goes through and this particular paper and talks about setting up an experiment or two and now we get really technical so um well just let this parse itself on the screen shows you a pulse circuit this is the philosophy and technique behind doing this whole thing and obviously they’ve managed to miniaturize all of this electronics down to where it now fits in these little fitness bands okay and then they go on to do a second experiment and this is just the first two pages of the whole article because you got to buy the whole article this is what they give you for free you’ll see the link down below if you’re really interested in this you can dive further so we’ve moved on from ECG and PPG measurements to determine the pulse wave transit time from that and using specialized circuitry we can analyze the signal difference to get the PW TT and work with it so we are talking about an electro plate type blood pressure bracelet bracelet that’s based on the principle of PW T T to measure blood pressure you got the blood pressure meter reading the ECG test less ppg and heart rate those are kind of the three things that are going on now in this new family of devices the PW TT principle trying again to make it talk about it in so many different ways that makes sense pulse wave transmission speed and blood pressure is directly related to high blood pressure pulse transmission fast otherwise slow I love Chinese English through the ECG signal and the pulse wave signal PPG can be obtained pulse wave transmission speed p WT t plus wave transmission time it’s actually pulse wave not gonna let the Chinese Aires slip through but their length their own unique languaging is okay we can figure it out anyway the the pulse the ECG and the PPG signaling can creat can get you the P P W TT and then from that calculate the blood pressure value so once again this is the eat CG signal that you see and a certain point in time can be registered when it finds that peak this is the optical thing coming through and at a certain time it can determine that bump in your capillary responds back to the sensor and it can measure the time difference between these two on your arm with that time difference it can then go and calculate your blood pressure when testing the ECG the right hand on the metal body on the bracelet do not touch the left arm like that okay you can’t rest your finger against it you can’t rest your thumb that’s why most of them have a button on the front see here’s the electrodes on the back and the other electrodes on the side that you squeezed some of them have it on the front much much much easier but some have it on the side either way could work as long as it bridges your heart the two electrodes at the bottom of the of the hand ring are in full contact with the left hand the three parts will form an electric circuit and the combined with the green photoelectric sensor test result calculate the blood pressure data all right we’re almost done one more document I wanted to go through oh gosh yeah it’s long but it really lays the total foundation for this so if you’re in school especially if you’re studying medical or health you might want to write a theme paper on this I think you could get an A mr. ticks I’ll help you with this I’m not gonna answer a lot of questions I don’t know much more than what I’m showing you but I mean this is good stuff you guys should seriously think about writing a dissertation well at least a home homework paper the emergence of mobile technology advances in these biosensors are to change the conventional healthcare system to facilitate systems that provide for mobile and individual centered health care systems mobile monitoring systems can provide continuous physiological data and better information regarding the general health of individuals I love good English for example such a mobile cardiac health monitoring system can reduce healthcare costs by disease prevention and enhancement of the quality of life with disease management ie this could be a health a preventative device that can be used in wellness by monitoring correctly your bio data and especially being able to get the ECG information the actual chart and send that off to your health care provider if you have some form of an incident that needs monitoring but happens so quickly they don’t have time to hook you up at the doctor’s office or get you into the hospital something like this might give you that preliminary data that could find something way more than just PW T T thing I mean when you look at all of the jiggles on that they all mean something in doctors and medical folks know what that is cardiologists valuable so if it’s reliable and if it’s accurate which it’s professing that it is because it’s actually giving you that electrical hook up directly to your heart then that data may be really really valuable so even if you don’t use it for blood pressure and heart rate the fact that you’ve got ECG output in the form of the chart that can be made into a PDF that can be emailed is amazing according to mobile device accordingly a mobile device is disclosed that determines a user’s cardiac health status by monitoring multiple key cardiovascular parameters and/or their index such as ECG heart rate cardiac output and blood pressure in a continuous and non invasive fashion for examples users can conveniently carry currently handheld mobile devices anywhere and conduct self monitoring when desired or necessary all the time or as needed when convenient and that’s what these medical devices often do that you you you know stuck on a belt or something like that they’ll measure you 24/7 for three weeks and then they dump all that data at the doctor’s office and work with it we’re talking about a little wrist kind of thing that can either give you spontaneous instant measurements over 30 seconds to two minutes or actually be able to do continuous measurement it’s storing that data that’s the difficult thing it can certainly generate it but you know anyway we digress the the home fitness bands are going to just basically give you your ECG heart rate and blood pressure information all right I’m not gonna read anymore let’s just let you skim this whole document because it’s giving you kind of a background pep talk about why this is good okay but we will get in here and says it is commonly accepted that the pulse wave transmit time can be registered as an index of arterial stiffness and has been employed as an indirect estimate of an indirect estimation of blood pressure P wtt can be measured at the time as the time interval between the R wave peak of ECG and the pulse wave arrival in the same cardiac cycle when ECG and pulse wave are simultaneously recorded PWD then it can be derived PWD TT was originally applied to the area blood pressure estimation and here’s a reference for all that and since then the researchers have studied the mechanism of the method and on and on so now you’re getting the background that has led up to the science of being able to do this with cuff less blood pressure estimation based on PT t t pulse transit time techniques right it’s actually pulse wave PW DT so forth and so on okay so bottom line what we are looking at in these new devices that Mister tix is all excited to be bringing you a bunch of different bands a bunch of different watches as this new technology emerges and becomes the de facto standard for say 2018 to 2019 in wearable devices what we’re looking at is the science behind how that works and why it is presumed estimated anticipated in hope to be relatively accurate and usable for collecting biometric data for your fitness and overall health alright thanks for watching this really if you stuck with it the whole way I applaud you hopefully you can use it somehow in in your life and it’ll be valuable to you in the meantime you now know what we’re talking about when you see something that says it’s got ECG plus PPG once again if you are interested in this particular band the t6 check the show notes down below for a buying link pick it up from Banquet are supporting partner on this watch and enjoy the new ECG PPG technology that’s coming out in this new form factor we’ll see you again soon..

How to Encrypt Files on WindowsIf you're looking for a simple way to keep files and folders private on your Windows computer, you have several options right in front of you. Microsoft Office Suite, you can use a built-in encryption feature to password-protect Office files, such as Word documents or PowerPoint presentations.Some Windows operating systems also come with Encrypting File System (EFS), which lets you encrypt any kind of file, as well as whole folders and subfolders. Note, however, that EFS is only available for Windows 10 Pro, Windows 7 Professional, Windows 7 Ultimate, Windows 7 Enterprise, Windows 8 Pro or Windows 8 Enterprise. Users with a Home edition of Windows will need to use either Office Suite encryption or a third-party solution, such as TrueCrypt, VeraCrypt or 7-Zip.To set up your Windows encryption, you'll want to follow these step-by-step instructions.Before you start altering your files, there are some tips you need to keep in mind.Anything can be decrypted if you're targeted by a savvy enough, or well-financed foe. You might want to find a paid solution if your files are truly valuable.Files encrypted using the below methods can still be deleted, so you might want to have a backup on a secondary location.If you lose your passwords, you've lost your files forever. So, again, keep an unencrypted backup on a physical drive somewhere safe where it won't be found.How to Encrypt Files Using Microsoft OfficeThis process encrypts individual files compatible with Microsoft Office applications such as Word, PowerPoint or Excel. Once you encrypt a file this way, you'll need to reopen it in Microsoft Office; you won't be able to open it in Google Docs, Adobe Reader or LibreOffice. These steps work for all up-to-date versions of Office, across Windows 7 and Windows 10.1. Open a Microsoft Office program and click Open Other Documents.2. Click Browse.3. Select a file you want to encrypt and click Open.4. Click the File tab at the top of the page.5. Click "Protect Document" on the left side.3. Select Encrypt with Password from the pop-up menu.4. Enter a password for the file. You'll be prompted to re-enter the same password, then click OK. After you exit this file, you'll have to enter the same password to reopen it. Be sure to store this password in a separate, safe place.How to Delete Temporary FilesYou're not quite done yet, though. One of the flaws with Microsoft Office's encryption is that unencrypted versions of recently opened files might still be stored in your computer's temporary memory. You'll want to go clear that out after you've encrypted a file.Click the Start button.2. Type "Disk Cleanup" into the text field and select Disk Cleanup.3. Wait for the loading bar to complete, it's calculating how many files it will be able to delete.4. After the window "Disk Cleanup for OS (C:) appears, check the box next to "Temporary files" (you may need to scroll down) and click OK.5. A new pop-up window will appear asking you to confirm the deletion. Click Delete Files.6. You’ll see a new pop-up window (pictured below) with a loading bar running as your files are deleted.Once it's finished, the window will disappear and the temporary files are gone.How to Encrypt Files on Windows using Encrypting File System (EFS)EFS works by letting you apply encryption to already-existing files or folders in your file system. You can still edit or modify these files or folders following the encryption process. With EFS you won't notice any change in the way you access your files; all you have to do is log in to your Windows account at startup and the files will be accessible. However, this means that you need to pick a strong, difficult-to-guess password for your Windows user account. Note: Step 7 is time-sensitive, so make sure to click the "Back up your file encryption key" prompt after confirming attribute changes in step 6. Missing that prompt means you'll need to start over again.1. Right-click on the file or folder you wish to encrypt.2. Click Properties selection at the bottom of the menu.3. Click Advanced under the General tab. This will bring up a second pop-up window entitled Advanced Attributes.4. Check "Encrypt contents to secure data."5. Click OK.6. Click Apply.7. Choose how extensive you want the encryption to be, click OK. You can choose to encrypt just that folder, or to encrypt all of the folder's subfolders and files. We recommend the latter. Whichever you choose, click that option and then press OK.8. Make sure to click the "Back up your file encryption key" pop-up message before it disappears. If you miss the pop-up message, you'll need to restart your machine and try again.The computer creates an encryption key using an encryption certificate provided by Microsoft. Now your file or folder is encrypted, you won't need a password to access it other than the password you use to sign into your Windows profile when you turn the computer on.How to Back Up Your Encryption KeyYou should back up that encryption key to a separate device, because if that key is ever lost or damaged, you won't be able to access your encrypted files. The easiest method is with an external USB drive, so plug one into your PC before starting.1. Click the option "Back up now (recommended)."2. Click Next.3. Click Next again.4. Check the box next to Password, enter your password twice and click Next.5. Click Browse.6. Navigate to a directory, such as a USB drive, name your encryption key and click save.7. Click Next.8. Click Finish.9. Click OK, now eject your USB drive (or wherever you stored the file) and keep it somewhere safe where you'll remember it.

An explanation of the Mandelbrot set for artistsFigure 0. The Mandelbrot set.You have seen this image before, or some variation of it. You might also know that it's called the Mandelbrot set and that Benoît Mandelbrot was the first to thoroughly study it in the late 1970s, though the image Mandelbrot first looked at might have looked more like this:Figure 1. A mock-up of what the Mandelbrot set might have looked out in a print-out from the 1970s. The reason for the flip will be explained in due time.You might also know that it has something to do with math. You might even know that it is an example of a fractal.But the problem with most explanations of the Mandelbrot set is that they seem to require advanced knowledge of calculus, plus the whole Greek alphabet. It is a very interesting image, and some artists are of course interested in incorporating it into their artwork.For this reason, I feel that it's necessary to have some explanation of it that is accessible to artists, who can be counted on to at least have a decent grasp of practical arithmetic and very basic algebra.The two key mathematical concepts to understand here are iteration and "complex" numbers. As iteration seems to be the easier concept to explain, that's what I will start with.IterationIteration is just the repeated application of a function, at each step after the initial step using the result of the previous step as the input. Depending on the function and the initial value, iteration can produce an infinite sequence of distinct numbers, or it can repeat the same numbers over and over again in some pattern.The iterated function that is of special interest to us here is this: multiply a number by itself and add a constant value. Algebraically, we can express this as z2+ c. Here, z2is z multiplied by itself and c is the constant value (if you remember your high school algebra, the choice of z rather than x may seem strange; it will be explained later on).In my opinion, the best choice for the starting value is z = c. However, technically, the correct choice is z = 0. But since 02= 0, all this does is add an unnecessary extra step to the calculations. It's not a big deal for our fast modern computers, but in 1995, details such as these made the difference between an image showing up on the screen in a matter of minutes rather than a matter of hours.Let's try c = 2. The function is then z2+ 2. The first iteration gives 22+ 2 = 6. The next iteration gives 62+ 2 = 38. And the next: 382+ 2 = 1446. Then 2090918, 4371938082726, 19113842599189892819591078, ... If you're following along with a pocket calculator, you've probably already gotten an overflow error.You can go a little farther with a scientific calculator, and farther still on a computer with a program like Maple or Mathematica. But you don't need a bigger calculator to see that these numbers just keep on getting bigger and bigger. A mathematician might say that the iteration escapes to positive infinity.Now let's try c = −2. The function is then z2− 2. The first iteration gives (−2)2− 2 = 2 (remember that a negative number times a negative number is a positive number). The next iteration gives 22− 2 = 2 again. The iteration is stuck on 2. A mathematician might say that the iteration attains a stable orbit. A regular person might say the iteration attains a boring orbit.We can express this concept of iteration a little more algebraically by using subscripts to keep track of how many iterations have been performed. Write z0= 0, and, for n a positive integer, zn= (zn − 1)2+ c.Using this notation with subscripts, we have z0= 0 and z1= c. Then z2is the first iteration for which we don't automatically know the value. So in the case of c = −2, we have z0= 0, z1= −2, z2= 2, z3= 2, z4= 2, z5= 2, etc.nzn10.2500000020.3125000030.3476562540.3708648750.3875407560.4001878370.4101503080.4182232790.42491070100.43054911......39910.4997500739920.4997501339930.4997501939940.4997502539950.4997503239960.4997503839970.4997504439980.4997505039990.4997505740000.49975063Table 1. A few iterations of zn= (zn − 1)2+ c, with z0= 0 and c = 0.25, to eight decimal places.Escaping to infinity or getting stuck on a particular number aren't the only two possibilities of iteration. Consider c = 1/4 = 0.25, so the function is then z2+ 1/4. The first iteration gives us 5/16 = 0.3125. Then 89/256 = 0.34765625, 24305/65536 = 0.3708648681640625, ... This is not going to escape to infinity.It might get stuck on 1/2 = 0.5. After all, 1/2 times 1/2 equals 1/4, and when you add 1/4 to that you're back to 1/2. That's actually not what happens either. After a thousand iterations, we have approximately 0.49900959. After two thousand iterations, we have approximately 0.49950258. And after four thousand iterations, approximately 0.49975063.Those numbers are from a spreadsheet program with a display precision of eight decimal digits (I can set more precision, if I want, but there are limitations). Carried far enough, this iterated function will take the computer to a point where the computer can't distinguish between 1/2 and the actual number. But with infinite precision, the iteration gets "arbitrarily" close to 1/2 without ever actually reaching it.We can still view this as an either-or: either the iterated function escapes to infinity or it doesn't. Take the real number line, with c = 0 at the center, painted in white, and change to black the pixels corresponding to c such that the iterated function z2+ c does not escape to infinity.So as to avoid getting the computer stuck in an infinite loop, we'll program the computer to keep track of how many steps in the iteration it has performed and instruct it to stop at a certain threshold and move on to the next point if it does not find evidence that the iteration escapes to infinity. Also, we'll instruct it to assume that the iteration has escaped to infinity if a step results in a value greater than 2.And so, by this process, we obtain a black line that goes from c = −2 to c = 1/4.Figure 2. The points on the real number line that attain stable orbits painted in black.Not the most exciting diagram. But we haven't yet ventured out to a wider universe of numbers.Complex numbersA "complex" number has a "real" part and an "imaginary" part. This whole terminology is unfortunate because it suggests that complex numbers are too complicated to be of any use, and that imaginary numbers have no basis in reality.But at least it allows me to draw a parallel to Impressionism in the visual arts. The term started out as an insult, but now critics and collectors have come to appreciate Impressionism, while mathematicians, scientists and engineers have recognized the usefulness of imaginary and complex numbers.The real part of a complex number is a real number, and the imaginary part of a complex number is an imaginary number.We can use real numbers to measure most solids, liquids and gases, which is what helps give these numbers a sense of reality. What the imaginary numbers measure are things which we have only recently come to understand, or are still struggling to understand, things like electricity and quantum particles.However, the context in which mathematicians first encountered imaginary numbers was in algebra.Real numbersA simple equation like x2− 1 = 0 has one obvious solution and one less obvious solution that should still be easy to come by. But if you have any difficulty whatsoever with this sample equation, you can make things easier by adding 1 to both sides, thus going from x2− 1 + 1 = 0 + 1 to just x2= 1.Then the obvious solution is x = 1. The less obvious solution is x = −1 (remember that a negative number times a negative number is a positive number). This is not the place to expound on the fundamental theorem of algebra, but there is one corollary of it that is of relevance here:If y is a positive real number, then the equation x2= y has two solutions, one positive (called the "principal" square root), the other negative.Imaginary numbersNow let's look at a very similar equation: x2+ 1 = 0, which we simplify by subtracting 1 from both sides to obtain x2= −1. Your calculator is not going to self-destruct because you ask it to compute the square root of a negative real number, but it will lock up in an error state and you'll have to press C or CE to do anything else.Confronted by equations of this kind long ago, one mathematician had the bright idea of assuming that a solution does exist and seeing what happens. Let's call this solution i. All we know about i at the moment is that i2= −1. The solution to x2+ 1 = 0 is x = i.That mathematician's contemporaries must have thought this was some kind of trickery or sophistry. We can imagine that an equation has as many solutions as we like, but that doesn't change the reality of an equation that has no real solutions, they would say.It was a slow process to realize that imaginary numbers do have a basis in reality and that they are part of a consistent number system, but the terminology stuck.If a positive real number has two square roots, does a negative real number also have two square roots? If we multiply i by −1, we get −i. Then (−i)2= (−1 × i)(−1 × i) = (−1)2(−i)2= 1 × −1 = −1. Hence x = −i is another solution to the equation.Take a real number b and multiply it by i to get bi (in general, mathematicians prefer not to use the multiplication sign unless it's absolutely necessary). That's all an imaginary number is: a real number multiplied by i.If y is a positive real number, then the equation x2= y has two solutions, one is √y and the other is −√y. And the equation x2= −y also has two solutions, one is i√y and the other is −i√y.Complex numbersI'm not going to go in-depth about the square roots of nonzero imaginary numbers. Suffice it to say that they are complex numbers, i.e., numbers with a nonzero real part and a nonzero imaginary part. For example, the square root of i is approximately is 0.7071 + 0.7071i.It's more important now to go over the arithmetic of complex numbers. If x and y are real numbers, yi is an imaginary number and x + yi is a complex number.Let's say v + wi is another complex number. Then (x + yi) + (v + wi) = (x + v) + (y + w)i. For example, (−3 + i) + (7 + 2i) = 4 + 3i. We can add a purely real or purely imaginary number to a complex number, just so long as we take care to put in zeroes at the appropriate spots so as to properly line up the real parts and the imaginary parts.To multiply complex numbers is not complicated at all. The FOIL mnemonic you might remember from high school is helpful here, but it winds up being a little more like FLOI: (v + wi)(x + yi) = (vx − wy) + (vy + wx)i. For example, (−3 + i)(7 + 2i) = (−21 − 2) + (7 − 6)i = −23 + i.As with the formula for the addition of complex numbers, the formula for the multiplication of complex numbers can also be used if one of the multiplicands is a purely real or purely imaginary number, so long as we take care to put in zeroes at the appropriate spots.The complex planeIf the real number line is horizontal and includes 0, and the imaginary number line also includes 0, it makes sense to make the imaginary number line vertical and attach it to the real number line so that they're perpendicular and their intersection is at 0.This is the basis for a diagram of the complex plane. Any complex number can be located in the complex plane by first identifying the real part on the real number axis and the imaginary part on the imaginary number axis and then drawing lines into the appropriate quadrant of the complex plane. Where those two lines meet is where the specific complex number is.Figure 3. A complex plane diagram.If you are looking at a print-out of this, feel free to try your hand at locating a few complex numbers on the complex plane with a pen or pencil. Don't worry about being precise; the point of the exercise is to understand the general idea.A few quick details of notationIt's not a rule, and it's certainly not written in stone, but mathematicians generally like to use x and y for purely real numbers. Then yi is a purely imaginary number and x + yi is a complex number.Mathematicians also like to use a single letter or symbol for anything that they have to write more than once. Even if it's just twice, but you have to admit that some of the things we're talking about here have been mentioned several times. So rather than writing x + yi over and over again, x + yi = z seems almost logical and natural.And z can still refer to a purely real number (in which case y = 0, and obviously yi = 0 also) or a purely imaginary number (in which case x = 0).The functions Re and Im come in handy: If z = x + yi, then the real part is Re(z) = x and the imaginary part divided by i is Im(z) = y. There are some very nice and fancy symbols for Re and Im, but depending on your computer's font, you may or may not be able to see them: ℜ and ℑ (if your computer's font doesn't have these symbols, you should be able to see them in Figure 3 above).Another one that comes in handy is the absolute value function. If −x is a negative number, then abs(−x) = x; for example, abs(−7) = 7. And if x = 0 or a positive number, then abs(x) = x.Measuring distance in the complex planeIt is easy to tell how far a particular purely real number x is from 0. If x is positive, then x is x away from 0. And if x is negative, then x is abs(x) away from 0. Since x = abs(x) when x is positive (or 0), we can just say that abs(x) tells us the distance of x from 0.Come to think of it, determining the distance of a purely imaginary number yi from 0 is also very easy. As we agree that y is a purely real number, it follows that both y and yi are the same distance away from 0, namely abs(y).With the right technique, this determination is also easy for a complex number x + yi with nonzero real part and nonzero imaginary part (meaning x ≠ 0, y ≠ 0 either). The right technique is to take advantage of the Pythagorean theorem, which you might remember from high school geometry.The ancient Greeks did not know about complex numbers, but they did know that in a triangle with a right (90-degree) angle, the longest side (the hypotenuse) is measured by the square root of the sum of the squares of the two other sides. If c is the longest side and a and b are the other two sides, then we have c = √(a2+ b2).This is relevant to a complex number x + yi, not purely real nor purely imaginary, because we can form a triangle with a right angle by drawing a line from 0 to x, then to x + yi and then back to 0. Here's an illustration for −4 + 3i:Figure 3a. A complex plane diagram with a black dot for the point −4 + 3i, and a light green triangle with that black dot, −4 and 0 as vertices.If you prefer, we can also form the triangle by drawing a line from 0 to yi, then to x + yi and then back to 0. Either way we have a triangle with a right angle and the longest side is the distance from 0 to x + yi. Which means that x + yi is √(x2+ y2) away from 0.And we don't have to worry about either x or y or both being negative, because they'll square to positive numbers anyway. So in our example we have (−4)2+ 32= 25, and √25 = 5. This means that −4 + 3i is 5 away from 0.I deliberately chose an example involving integers only, in order to keep things simple; which square integers can be expressed as a sum of two other square integers gets into algebraic number theory, which is very interesting in its own right but is beyond the scope of this page. Already this subsection might seem like a digression from the topic of the Mandelbrot set.We can also apply this Pythagorean principle to measuring the distance between two complex numbers x + yi and v + wi such that x ≠ v and y ≠ w. Simply compute (x + yi) − (v + wi) or (v + wi) − (x + yi) (doesn't matter which) and then compute the distance of that number from 0.Iterations on imaginary and complex numbersWe're now ready to start putting all this together. We'll start by computing a few steps of the iteration of z2+ c with c = 2i. Obviously (2i)2= −4, so the first step of the iteration gives us −4 + 2i. Then 12 − 14i, −52 − 334i, −108852 + 34738i.These numbers are not escaping to positive infinity, but they are getting very far away from 0 very fast, and in a very practical sense, they're growing too large for our computers to handle. We could say that the iteration is escaping to a "directionless" infinity.We could program the computer to consider that the iteration has escaped to some infinity if either abs(Re(z)) > 2 or abs(Im(z)) > 2, or both, and thus move on to the next point.Now let's try c = i. Remember that i2= −1, so the first step of the iteration here gives us −1 + i. Next we have (−1 + i)2= −2i, so adding i gets us to −i. Take a moment to verify these calculations with the multiplication formula given a few paragraphs earlier.Remember that negative real numbers have two square roots: one is positive imaginary, the other is negative imaginary. Thus i2= (−i)2= −1. Therefore the next step in the iteration brings us back to... −1 + i.So the iteration bounces back and forth between two distinct values. This is a stable orbit. Pretty much the same thing happens with c = −i.Can we assume that every point in a straight line between i and −i attains a stable orbit? That is, that if Re(c) = 0 and −1 ≤ Im(c) ≤ 1, then the iteration of z2+ c does not escape to some infinity? That would be a wrong assumption, but let's run with it for now.If we paint black on the real number line and the imaginary number line those points that don't escape to infinity, using the assumption mentioned just now, we get a cross:Figure 4. The points on the real number line that attain stable orbits painted in black together with the points on the imaginary number line which we have guessed also attain stable orbit.Not the most interesting thing, but we haven't looked at complex values of c yet. If we were to make assumptions about each vertical line, like we did with Re(c) = 0, it would still take us a long time to go through. So at this point, it would be perfectly reasonable to guess that the diagram we've been working on should look something like this:Figure 5. Our guess as to what the diagram will look like once we take the complex numbers into account.But that's based on a few faulty assumptions, the first one being that all purely imaginary c between i and −i attain stable orbits under the iteration. It's not hard to find c to prove that assumption wrong. For example, c = 9i/10, which takes less than ten iterations to start getting far away from 0.As humans, we can grow impatient at having to do a bunch of calculations, whereas a computer will not complain about the drudgery of a million calculations, provided it has been properly programmed.To produce the diagram we've been talking about on a screen with HD 1080P resolution requires looking at more than two million distinct values of c, which means at least two million calculations: just one or two for the rapidly escaping points, and as many as the iteration maximum we've set for the stable orbit points. And so, with the proper programming, we should obtain the following diagram:Figure 6. The diagram we've been getting at.To have some semblance of mathematical rigor, I should explain that this diagram shows us the Mandelbrot set but is not actually the set. The Mandelbrot set is the set of all points c that attain a stable orbit under the iteration of the function z2+ c initialized with z = 0 (or c).Although Figure 6 is way more correct than Figure 5, it would lead us to believe that the various little dots surrounding the main mass of points c are not pathwise connected, but they in fact are. Well, at least the ones that can be seen in Figure 6, I'll delve a little bit more into this question a little later on.But perhaps now is the proper time to explain the flip. Mandelbrot and his colleagues were studying the iteration of z2− c rather than z2+ c. But all this does is flip the diagram along the imaginary number axis (which is vertical).Getting your computer to draw the Mandelbrot setBack in 1995, I copied a simple program off a book. I made a few modifications to the program, some which were absolutely necessary for the program to run on the setup available to me at the time, some were purely stylistic and affecting only the appearance of the source code, and a few were to optimize performance.To show you how fast computers have gotten in the past twenty years, I'm going to tell you about a couple of the performance optimizations. One of those modifications turned out to be utterly unnecessary for today's computers.I'm fairly certain the program as written in the book did not take advantage of the symmetry of the Mandelbrot set (e.g., if you know that c = a + bi escapes, then you also know that c = a − bi does too). Presumably that was to allow readers to modify the program to draw an area of the Mandelbrot set diagram that might not necessarily be symmetrical.So I modified the program to take advantage of the symmetry and thus cut the calculation time in half. Thus the program worked in two vertical lines from left to right, with one line starting at the top of the screen and the other at the bottom, and meeting each other in the middle of the screen.I also added a line to the program instructing the computer to place a white or red dot (I don't remember which color it was) at the point c before doing the iterative calculations and coloring the point according to those calculations. The effect of this was to cause two white or red dots to move across the screen as the computer drew the Mandelbrot set.This way, as the computer moved through several consecutive stable orbit points, you had some indication that it was doing something, that it hadn't crashed. Well, in 2005, I dug out the program with my modifications and ran it: the computer drew the Mandelbrot set so fast the moving dots were completely unnecessary, I couldn't even see them.While I can pat myself on the back for my ingenuity on the modifications, the fact is that it would have taken a lot of work to turn the program into a full-fledged fractal program. I briefly considered working on just such a program, but fortunately I became aware of FractInt before I took on that massive endeavor.The WinFract Basic Options dialog box, accessible through Fractals > Basic Options...FractInt and WinFractI'm not sure at what point I became aware of FractInt. It was a DOS program that was already available in 1995, and which back then could be obtained by sending a check or money order (for a modest shipping and handling fee) to some address I don't remember, and receiving a 3.5" floppy disk to install the program. They would also send an iron-on so you could put a fractal on a T-shirt.If I still had that floppy disk, its only value would be as a collectible. Nowadays you can just download FractInt (and WinFract) from the Internet like just about every other program. Just be sure to get it straight from fractint.org (the link should open in a new window or tab) so it comes by itself, not bundled with any obnoxious adware or whatever.As late as 2010 I got the impression that DOS was still a fundamental structure underlying the Windows operating system. I don't think that's the case anymore with Windows 8.1, though the DOS command line is still available. DOS is essentially a relic, as most of its users have moved on to Windows.WinFract has all the functionality of FractInt, but with all graphical user interface niceties we have come to take for granted in Windows. As much as I like keyboard shortcuts, I've really come to appreciate being able to use the mouse to point at things on the screen.In regards to fractals, the mouse is especially handy for telling the program where to zoom in. As it turns out, FractInt has had mouse support from fairly early on. Though I do wish WinFract had more keyboard shortcuts.The big downside to FractInt and WinFract right now (as of 2016) is that neither of them can run in Windows 7, 8 or 10. It needs to be ported to work with 64-bit processors. Even a simplistic recompile that does not necessarily take advantage of the computer's 64-bit capabilities would be a major undertaking, and is probably outside the skill set of the vast majority of people reading this page.Fortunately for me, I still have access to some older computers running Windows XP. WinFract runs like a dream on one of those, with speeds that would have astounded me in 1995. Most of the images on this page were generated by WinFract running on that computer. Some of them I touched up in Adobe Photoshop Elements 8 running on my Windows 8.1 computer at home.ChaosProI don't know about Windows 7 or 10, but I can tell you for a fact that ChaosPro 4.0 does run in Windows 8.1. And it's freeware, just like FractInt and WinFract. But be sure to get it straight from chaospro.de (the link should open in a new window or tab).It seems that ChaosPro can do everything that FractInt and WinFract can, and a lot more. But ChaosPro is geared more towards 3-dimensional images that you can move on a 2-dimensional screen. Along with all that power comes a very steep learning curve. It took me several minutes to figure out how to make it just draw the standard Mandelbrot set (which is the default fractal on both FractInt and WinFract).To draw the Mandelbrot set in ChaosPro, the menu command is File > Fractal > Escape Time, or you can use the keyboard shortcut Ctrl-4. I have yet to figure out how to get it to draw the other 2-dimensional fractals that I know FractInt is capable of and which I believe ChaosPro can do as well.Fractal programs for other platformsFractInt has been ported to Linux, and I think also to Mac OS X. There's also FRAX for iPhones and iPads; it is said to be the best fractal program for touchscreen devices. I am also aware of fractal apps for other smartphones, but I would be wary of whether they perform the calculations on your device or send them off to a server and then retrieve the results; look for phrases like "data transfer rates may apply."What's up with the colors?Even if you don't remember, you've surely seen a few different color variations for the Mandelbrot set. Not even the correspondence of black to points that attain a stable orbit is standard as much as that may seem to be the case: you may have seen illustrations in which the stable orbit points are colored blue or red or some other color.Figure 7. The Mandelbrot set with the default colors in ChaosPro 4.0.I personally prefer black for the stable orbit points, and I think lots of other people do, too. The boundary between those points and the points that escape to infinity, that's what mathematicians care about. For that purpose, black and white is sufficient, and for mathematicians, sufficient is perfect.But for computer graphics enthusiasts, black and white does not even begin to show off the capabilities of their screens. Thus they have come up with various schemes to illustrate the Mandelbrot set with a greater variety of colors.One option to bring in more colors is to color the escaping points according to how fast the escape is. Points that escape very quickly, or immediately, like c = 2, could be colored a dark blue. Points that escape slowly, like c = 0.251 (the iteration takes 96 steps to go above 2), could be colored white (remember, for these points, you want a light color to constrast with the black of the stable orbit points).Now, some people complain that this produces some ugly banding. There are ways to still use the speed of escape to determine the color yet create smooth gradients among the escaping points. I don't care much for this one way or the other, I am of the opinion that as long as the coloring algorithm does not distract from the main interest (the demarcation between the stable orbit points and the escaping points), it is quite unimportant.The way FractInt and WinFract do colors is by means of "color maps." default.map is the default color map, which is a fairly straightforward use of the colors that were available on the old VGA displays.Both FractInt and WinFract come with about twenty other color maps besides the default map, plus it looks like you can also design your own color maps. But in my experience, other color maps cause glitches with saving images, so you may have to use the Alt-PrintScreen kludge.Zooming in on the Mandelbrot setThere are more wonders of the Mandelbrot set to see when we zoom in. WinFract has the Zoom In box selected by default: you simply click and drag to make a rectangle (which you can reposition if you want) and hit Enter.Figure 8a. A zoom of the Mandelbrot set. I forgot to write the coordinates for this one.Depending on the program and how close you zoom in, you might need to adjust the number of iterations. If things start looking blocky, fuzzy or just generally not sharp, the number of iterations might need to be increased.The WinFract default is a maximum of 150 iterations per point, which is just fine for a drawing of the whole set on an HD screen, but can be problematic even for modest zooms.To increase the iteration maximum, the menu command is Fractals > Basic Options... This brings up a dialog box in which you can type in a number for "Maximum Iterations." A new value of 1,000 might be sufficient if you have only zoomed in a couple of times.The maximum number of maximum iterations per point in WinFract is 2,147,483,647. This number was not pulled out of a hat: it is the largest number a computer can hold in memory as a signed 32-bit integer. Presumably if WinFract was ported for 64-bit Windows this maximum could be boosted to 9,223,372,036,854,775,807.However, even with our faster computers, the maximum iteration maximum is probably overkill if you're not looking to test how close you can zoom in. I suggest raising the maximum iteration maximum in steps of 1,000 or 10,000 as you zoom in.Figure 8b. A closer zoom of the Mandelbrot set.As you zoom in, you should notice small copies of the whole big shape. And these copies are in turn surrounded by yet smaller copies of the whole shape. Mathematicians say the shape is "self-similar." Self-similarity is one of the defining characteristics of a fractal.Long before computers made it practical to draw fractals like the Mandelbrot set, a few mathematicians were able to determine certain important facts about these fractals, but many questions remain. For instance, are all the little copies of the overall shape pathwise connected?To state this in a rigorous manner, we can say that from any point c such that the iteration of z2+ c does not escape to infinity, to any other point with the same property, it is possible to draw a path to connect those two points such that all points on the path also have that property. Or at least, we believe that it is possible to draw such a path.For example, chart a path from 0 to i without leaving the Mandelbrot set. As we already saw, 9i/10 is not in the Mandelbrot set, and neither are a lot of points surrounding it, so our path would have to avoid it. The final stretch of the path is especially tricky, and you'd probably want the diagram displayed in black and white in order to see the path.In many cases, the connecting paths will be very narrow and require a lot more iteration to reveal them than your computer is capable of. Also, many of those connecting paths can be incredibly long and tortuous.Figure 8c. A still closer zoom of the Mandelbrot set.There are also some smooth, straight line paths between little copies, such as those along the real axis. In the following image, you can see the little copy at c = −1.75, and maybe you can also see a couple other smaller copies lined up horizontally:Figure 9. A little copy of the overall shape at c = −1.75.Can the perimeter of the Mandelbrot set can be measured, like the perimeter of a square or a circle or an oval? If a square has a side measuring x, then obviously the perimeter is just 4x.The perimeter of a circle (its circumference) can be calculated with a formula that involves π, which is approximately 3.14159. It's a little more involved to calculate the circumference of the oval shown in Figure 5, but it's still quite doable.Well, for the Mandelbrot set, we can't calculate its perimeter, but we know what it is: it is positive infinity! Though I suppose that assumes the whole set is pathwise connected.