Shade Circles Like This Not Like This: Fill & Download for Free

Interesting question. It suggests a considered and intellectual approach to what, in most people, is an instinctive reaction (and a mysterious one). You say you don't 'respond' to them -- but it's not like you don't see or hear them. You became aware that your experience of other people is not the norm.It suggests that you don't 'feel' your way into people. Instead of sensing their inner world through the signals that your body (your embodied unconscious) picks up from their body, you apply yourself to 'reading' them like it's a practiced skill ("I'm not bad").There are two groups of people who lack empathy: sociopaths/narcissists/psychopaths, and people on the spectrum. A sociopath might be curious enough to ask this but wouldn't be troubled by it -- would see it as a strength. So you might have shades of autism or sensory processing disorder. There's nothing wrong with that, or you -- it just means that you're a rectangle in a world built for circles. There are techniques you can learn to keep yourself rolling along.Another possibility is dissociation, usually because of a trauma in early childhood. Trauma happens whenever an experience overwhelms your brain. Instead of processing it, the brain pushes it into a dark corner and acts like it never happened. But as your brain tells one story, the trauma tells another, secret story that sends out weird signals and messes up the main narrative (also known as 'your life'). You move through a gray zone, numbed out and disconnected; you overreact to some things and underreact to others. In that case, you need to find a capable therapist who can help you dig up the trauma, see it in daylight, and put it where it belongs.

When you can't figure out how to get started, ask yourself questions to make sure you understand the situation thoroughly. Frequently, just by explaining in detail what the problem actually says, you figure out the answer.What are we looking for?The radius of circle C3. Call it "r".What information do we have?The radii of circles C1 and C2, "s" and "t".Anything else?The picture.What about it?The circles are all tangent. The lines are parallel to each other and tangent to the circles.Does a solution exist?Of course. Just look at the picture.But what if the picture is deceptive? Does a solution exist for all values of s and t?I don't know. Why wouldn't it?We'll come back to it. If you were going to put this together with actual circles and lines, where would you start?With circles C1 and C2.Why?Because I already know their sizes.Forget their sizes. Suppose we just want something that looks roughly like the diagram, without the sizes matching any particular specification. Where would you start?Well, everything fits inside the parallel lines, so I'd start with them, and then the big circle of radius r fits inside.Okay. What next?Let's put in one of the smaller circles.Where does it go?On bottom. Tangent to the big circle and tangent to the bottom line.How do you get two circles to be tangent to each other?Well, in general it looks like this:so one thing we could say is that their centers and their point of tangency lie on a line.So, go back to your previous drawing...We'll pick a point on the edge of the big circle. The center of the little circle lies on the line between the center of the big circle and that point.Where on that line?There are lots of possible circles. I guess they look like this:But only one of those is right.Yeah. The circles get smaller until at some point they become tangent to the bottom line, then after that they get too small. We just one the one correct circle like this:What did this new circle depend on?On the point I chose on the edge of the big circle. So there could have been other choices if I had done it differently.What does that look like?Let's draw some different possibilities in.The blue circles represent possibilities tangent on the bottom. The red circles are possibilities tangent on the top.That's a little intimidating. What does a solution look like?Each blue circle has one red circle that's just tangent to it and vice versa, like this:Do all the blue circles have a red partner?No, actually. Some of them are too small. They hide underneath the big circle and never pop out far enough to touch any of the red circles. I guess that means the problem doesn't even have a solution for some values of s and t.Okay. Supposing it does have a solution, is the solution unique?Yes. There are many ways to draw in the red and blue circles in our diagram. However, each pair of one red/one blue has a unique ratio of sizes of those two circles, if we assume without loss of generality that the blue circle is bigger. For a given value of s and t, the ratio of s to t tells us which blue and red circles we want, so assuming a solution exists, it's unique.Good. Now we understand the problem. What are some features of the situation? Perhaps something we've already pointed out.Well, the circles are tangent, and we've said that that means their centers and points of tangent lie on lines. We can draw that in.A triangle appears!Yes. And the lengths of the sides just depend on the radii. Also, trigonometry gives us a relation between those lengths. It depends on the angles of the triangle, though.Okay. What else do you know about it?I'm not sure.It's often helpful to go back to where you outlined all the information given and see if there's something you haven't yet used. What given information isn't used in simply identifying a triangle?The fact that the lines are parallel. There would be a triangle anyway if they weren't.And what do you know about parallel lines?One thing is that they're always the same distance apart. Another is the transversals cross them at equal angles.Are either of those facts helpful here?Maybe... wait, what if we use the "same distance apart" feature to draw some lines like this:What's that?The green line and the purple lines both span the same vertical distance between the parallel lines. Also, they're all made out of the radii of the circles, so it lets us relate the radii to each other.What's the equation?I'm not sure. It depends on the angle of the slant of that middle purple piece.Notice anything about it?Wait a second, it's the same as one of the sides of the triangle. That means the angles in the triangle are related to the angle of the slant of the that middle piece somehow.Are we there yet?Not quite, but we're getting somewhere. Hey, we know a little about the vertical stretch of that middle piece. What about its horizontal stretch?What about it?Its horizontal extent also depends on how slanted it is.How can we find something else out about it?I don't know.We'll need to use the information we have.We're working with the radii of the circles, so we'll want to use those. Oh, now I see a way those radii relate to each other and to the horizontal slant. We look at these right triangles:Each shaded triangle has a certain horizontal extent that depends on the length of its hypotenuse and the size of its skinny angle. However, the hypotenuses are just radii of the circles.When we subtract the two horizontal extents of the triangles, we get the horizontal extent of the middle purple segment.Now do we have enough to put it all together?I think so. We have the first triangle we found, with goes between the three centers of the circles. That gives us a relation between the radii of the circles, but only if we know the angles involved.Then we have this purple line/green line thing. That gives us a second relation between the radii. If it used the same angle, we'd have two equation and two unknowns, meaning we could do algebra to finish up. Unfortunately, it depends on the angle of slope of the middle purple line, which is a new angle entirely. To finish things off we need a relation between the two angles from our first two approaches.That relation between angles is supplied by looking at the horizontal extent of the purple slope. On the one hand, this depends on the slope of the purple segment, but on the other hand it depends on the two angles of the right triangles we drew, which together make up an angle from our original triangle between the three circles' centers. Putting that all together should be enough information to solve the problem.Okay then. Go ahead and do it.

Because two words: road trip! Our skulls are different because we took a road trip. I had better explain. But we’re going to have to go back to the beginning. Our beginning. The beginning of the homo sapiens.Long, long ago, in a land far, far, away, our fathers had a home.That’s the cradle of life. Modern science almost unanimously agrees that we started there. That is also where the fabled Garden of Eden probably was. The Biblical account in the book of Genesis describes it this way:Genesis chapter 2 (NIV)5 Now no shrub had yet appeared on the earth and no plant had yet sprung up, for the Lord God had not sent rain on the earth and there was no one to work the ground, 6 but streams came up from the earth and watered the whole surface of the ground.10 A river watering the garden flowed from Eden; from there it was separated into four headwaters. 11 The name of the first is the Pishon; it winds through the entire land of Havilah, where there is gold. 12 (The gold of that land is good; aromatic resin and onyx are also there.) 13 The name of the second river is the Gihon; it winds through the entire land of Cush. 14 The name of the third river is the Tigris; it runs along the east side of Ashur. And the fourth river is the Euphrates.We got our start in a hot and dry place; but we were built for it. Hot, dry climates favor long, thin frames, because these frames would have a higher body surface area to mass ratio, enabling the body to lose heat faster. So that is how we were built. And heavy sunlight favors dark skin. So this is what we probably looked like.The same principles applied to our faces as well. Our skulls were narrow in portrait (from the front/coronal plane) and broad in profile (from the side/sagittal plane).And so we looked like this.Some of the greatest art in the history of mankind is art themed on Genesis (thank God!). And some of the worst art in the history of mankind is art themed on Genesis (Why, God, WHY?). Like this painting. The first homo sapiens looked nothing like this. (Not that the Bible says they did, for that matter. This is all the artist’s lack of informed imagination.)The first of us more likely looked like this.Anyways, we left home. We moved out, along the path of least resistance.When I say moved, I mean a process of gradual relocation and resettlement. We moved slowly. We fought wars with homo neanderthalensis, our more brutish cousin who still occupied the neighboring regions. We beat them at hunting. We won, we lost, we won again. We gained ground. Entire generations lived and died between moves that happened sporadically and gradually.Over time, we moved from Africa into Eurasia.Some of us went into the Indian subcontinent. We were still dark-skinned, with the skulls of our negroid ancestors. But unlike the hot grass of Africa, the wet, hot, and leafy tropical climate of India required a different set of features.We gained a tendency to store fat. Our skin tone lightened a bit. And our facial skeleton smoothed out a little. Since the overhead sun was no longer that strong, our hairlines slipped upward a little. We developed more prominent foreheads. Our faces became less narrow; and broadened out a little.We settled into the Indian subcontinent and then went island hopping on a Pacific honeymoon.Some of us spread out into the vast steppes of Eurasia. Endless plains of grass. Bitter cold. Overcast skies. Snow in the winter.So we developed a different set of features. Since the sunlight wasn’t that strong here, our skin tone further lightened out. To deal with the harsh wind that blew almost continually, our facial skeletons decreased any prominence. Our noses grew flatter to decrease exposure to cold.Facial hair increased the risk of condensation and ice forming on our faces, so we lost most of our facial hair. To deal with the harsh sunlight, we developed a set of epicanthal folds that reduced the glare from snow and ice. The folds gave our eyes an almond shape.As we moved north, the climate intensified. It grew colder, windier, harsher.So these new features (developed to enable us to adapt to the environment) increased in intensity.Our epicanthal folds got more prominent to the point we were looking out at the world through near-slits (not that we got any less efficient or alert as hunters). Our noses flattened further. But the harsh sunlight was incentive for our skin tone to darken a little one again. We didn’t get darker as much as we got redder, more sunburned. We began to store more fat in preparation for the long lean winters.And then we began to cross continents. We crossed the Arctic circle. And we crossed into America.We retained our original features in both places.In the Arctic.And in America.As we moved into the American mainland, away from the cold and closer to the equator, we grew leaner once more. Our faces grew more angular. The epicanthal folds around our eyes grew less prominent. Our skin turned a smooth shade of nut brown.That’s right. We got really good-looking.Later immigrants to America certainly seemed to think so. Context matters, of course. The later immigrants were essentially desperate refugees or glorified bandits, and had pale complexions, like the underbelly of a fish, with minuscule marks of pink and tan on close inspection. Their ears, noses and cheekbones stuck out at conspicuous angles. They were fleeing poverty and violence, and came a long way over the water. So when they reached America, this was what they had to say about us.They were well-built, with good bodies and handsome features…And when they met the Americans at Rhode Island (the Naragansetts) they described us thus:This is the finest looking tribe, and the handsomest in their costumes, that we found in our voyage. They exceed us in size, and they are of a very fair complexion; some of them incline more to a white and others to a tawny color. Their faces are sharp, their hair long and black, upon the adorning of which they bestow great pains; their eyes are black and sharp, their expressions mild and pleasant greatly resembling the antique. I say nothing to your Majesty of the other parts of the body, which are all in good proportion, and such as belong to well-formed men. Their women are of the same form and beauty, very graceful, of fine countenance, and of pleasing appearance and manners and modesty. ...They are very generous, giving away whatever they have. We formed a great friendship with them…The people making these remarks were some of us who made a side trip via Eurasia. We had gone west into Europe.Europe was north of Africa, and further from the equator, so it was colder. We grew fairer. And we met a dying ancestor of ours: homo neanderthalensis. They existed in a belt north of our original home, but they were present in higher concentrations here. And so we began to compete. We fought a lot. Then we f**ked (a little).Because we selectively slept with their women who resembled us the most, our gene pool was slightly diluted. So we still looked like us for the most part, except for the fact that we inherited the neanderthal’s most prominent features: their prominent brows and jutting chin.So when a lot of us (on the left) mated with a few of these (on the right) a new face shape emerged. Our chin grew and supraorbital ridges (above the eyes) grew more prominent. Our faces grew less narrow, so our mouths protruded less, causing our teeth to become more crowded together.And we began to look like this.As we moved up north in Europe, into colder and harsher climates, we retained those features but gained modifications for the climate. Like epicanthal folds (just like we had in north Asia) to hood our eyes. And thinner lips.Note: So when Hitler suggested that the ‘Nordic’ look represented the most developed of all races, the Master Race, he was the opposite of correct. If anything, the “master race” got its signature face shape by breeding with the ‘lowest’: a subspecies just below us. So much for that idiotic theory of racial purity that helped kill millions of ‘undesirables’. (I am not denigrating Scandinavians, who themselves are very nice people and don’t particularly think too much of their looks. And science would tell you that interbreeding results in hybrid vigor.)Whew. And that is the larger reason for the broad variation in skull shapes.As for individual variations, there is one reason that produces consistent results: growth hormone. This hormone induces growth of bones, both in the limbs and spine, as well as the face. Now, the bones of the limbs and spine close around puberty, but the bones of the face don’t. They can grow further if plied with more growth hormone.So if a person’s testosterone/growth hormone levels were consistently high in childhood, or rise in adulthood, after puberty (they cannot grow taller): their skull shape will change. The brow and chin will become more prominent. This is called acromegaly if it happens after adulthood is attained.Observe the differences below (normal to the left, acromegalic to the right).The skull does increase in size somewhat, and some features become more prominent. This happens gradually over time, as in the case of this man with acromegaly who graciously allowed himself to be photographed over the years.The rise in growth hormone, post-puberty, can happen for any number of reasons. It could be due to a tumor secreting growth hormone. It could be due to a rise in testosterone, which activates growth hormone. Athletes often tend to have prominent chins because of their naturally higher testosterone levels. People who abuse anabolic steroids for a very long time tend to gain similar features to a smaller extent.And then there’s genetics at the individual level. If you really want to know why your jaw sticks out more than your friend’s jaw, despite the fact that you are both of the same race, and have the same hormonal history, do what every male does when he has no clue and Google will not give him the answer: go ask your mom, moms know everything.Just remember this: the differences in skull shapes mean nothing. The shape and size of your skull does not determine your intelligence or your ability to do well in life. That is science. The opposite of this is racism. (It may call itself phrenology or Nazism, but it is racism masquerading as pseudoscience.)They say beauty is in the eyes of the beholder. But I’ve been the opposite of a romantic all my life. I’m utilitarian. Beauty, for me, is in the hands of the user. So I look at these two faces below and see that they are profoundly different. But I also know that both these faces were carefully evolved via a long, slow, and painful process of experimentation and elimination, to bring them to the point where they would do the most important thing anything could do: their job.So for an unromantic functionally-oriented person like me, both these faces, and every other face I’ve used in this answer of mine, are equally beautiful. Anyone who thinks otherwise is entitled to their own opinion.(Just kidding. Anyone who thinks otherwise is wrong. Period.)P.S. I’m deliberately omitting obscure practices like artificial cranial deformation that are incapable of being transmitted down generations via epigenetics. Those are obviously irrelevant to this question. And I’m omitting finer details on interbreeding because this is an entry on Quora, not the Encyclopedia f**king Britannica, and because I really need to pee. Excuse me.Advertising is a fundamentally bad habit, but I don’t smoke, or drink, or use, so I allow myself this one vice: if you liked this answer, you may like these as well.Asher Nitin's answer to How does Clark Kent get his company medical checkup without the doctors finding out he is Superman?Asher Nitin's answer to How do I form more connections in my brain so I can be a livelier conversationalist and a quicker writer?