Selection Guide For Round Poles And Square Poles: Fill & Download for Free

Alright, I think it’s been long enough, and it’s time to do another one of these! This time, we’ll be dissecting the cockpit of the Mirage 2000. And, as with my other answers…What do all the controls in an airplane's cockpit do? https://www.quora.com/What-do-all-the-controls-in-a-fighter-jets-cockpit-do https://www.quora.com/What-do-all-of-the-controls-in-an-F16-fighter-jets-cockpit-do… we’ll need to pick a specific Mirage 2000 to dissect. For this answer, I’ll be covering the Mirage 2000C, as flown since 1982. The “C” stands for chasseur (“fighter” in French).The Mirage 2000C is a French fourth-generation jet fighter. (“Fourth-generation” broadly meaning it has advanced air-to-air missiles, sophisticated radar display, digital mission and targeting computer, and more.) It began life as a lightweight fighter but has since evolved into a capable multirole aircraft with both air-to-air and air-to-ground capability.Let’s once again start by introducing you to the systems that these cockpit switches will control.Engine: The Mirage 2000C is powered by a single SNECMA M53-P2 engine, which is an afterburning turbofan. Its simple single-spool design is unique: Unlike most jet engines, which use separate spools for the fan and the compressor, the M53 has one spool, attached to which are the fan, the compressors, and the exhaust turbine. The engine is started by the battery.For stable operation in supersonic flight, the nose of the aircraft has a movable shock cone (souris), that retracts inward after exceeding the speed of sound, to control the location of the supersonic shock wave within the engine air intake.The intake has automatically-controlled scoops (pelles) that open to increase airflow into the engine during high-angle-of-attack maneuvering, where airflow would normally not be sufficient for continuous engine operation.The engine is controlled by a digital engine computer that manages fuel flow, nozzle position, and afterburner, to produce commanded engine power. In the event the main engine computer fails, there is a much simpler secondary engine computer (secours calculateur) that can keep the engine running at a reduced thrust level.In the event both computers fail, a secondary “emergency fuel” (secours carburant) mode is available, isolating RPM and nozzle control from fuel control.Fuel system: The Mirage 2000C has six fuel tanks split into two groups, the left group and the right group. The fuel tanks in each group are the wing tank, the forward tank, and the feeder tank. In addition, there is a center tank just aft of the cockpit. The Mirage 2000C can also mount up to three external fuel tanks for additional fuel, and has an aerial refueling probe ahead of the cockpit for in-flight refueling capability.Environmental control system: The ECS performs heating and air conditioning to automatically control air temperature and pressurization. The Mirage 2000C also has onboard liquid oxygen to provide oxygen to the pilot’s mask.Electrical system: The Mirage 2000C has AC and DC power systems. Primary AC power comes from two alternators, connected to the engine. The alternators are connected to two transformer-regulators providing primary DC power. If engine power is not available, a battery provides DC power, and it is connected to a power converter to provide AC power. In addition, the flight computer has a dedicated power converter.Flight controls: The Mirage 2000 is a delta-wing aircraft with no horizontal stabilizer. At the trailing end of the delta wing are four elevons, that provide both pitch and roll control. The vertical stabilizer has a rudder. In addition, the wings have automatically-controlled slats (becs) on the leading edges.The Mirage 2000C has a fully-digital fly-by-wire (FBW) control system. The FBW system is “full authority” — in other words, the airplane is not controllable if the FBW system fails. Therefore it is granted extra failsafes. The FBW system has four main control computers and a fifth emergency computer. The FBW system operates in one of three different modes: air-to-air mode (allowing full maneuverability), air-to-ground mode (reducing maneuverability when heavy ordnance is loaded onto the wings), and emergency mode (a degraded functionality mode used when the main FBW system fails).The Mirage 2000C also has airbrakes above and below each wing to slow the aircraft down. On the ground, a nosewheel steering system links the rudder controls to the nosewheel, allowing it to turn the aircraft on the ground.The Mirage 2000C can also mount a drag chute or arrestor tail hook. Both are used to slow the aircraft down in an emergency after landing.Hydraulic system: The Mirage 2000C has two independent hydraulic systems, called system 1 and system 2. Each system has one engine-driven main pump and one electrically-powered reserve pump. The various hydraulically-powered devices on the aircraft are distributed across the two systems.Each hydraulic system also includes a hydro-alternator that generates electrical power for the #1 and #2 fly-by-wire channels, allowing continued FBW operation in the event of an electrical failure.Landing gear: The Mirage 2000C has tricycle landing gear (nose wheel and two main wheels). The main wheels each have carbon disc brakes and are connected to an anti-skid system. There is also a parking brake.Autopilot: The Mirage 2000C’s autopilot is closely linked to the fly-by-wire system. The aircraft is designed to be flown on autopilot most of the time (except in combat, of course). The autopilot operates in one of a few different modes: attitude hold (holds the commanded pitch and heading or bank), altitude hold (holds commanded heading and altitude), and approach mode (flies an approach to a runway).Pitot-static system: The nose of the aircraft contains an air data probe consisting of a pitot and static port, which are used to power the barometric instruments like altimeter, airspeed indicator, and vertical speed indicator.Lighting system: The Mirage 2000C has both interior and exterior lights. Interior lights illuminate the cockpit and panel text. Exterior lights include navigation and formation lights, anti-collision lights, landing lights, and a police light.Communications system: The Mirage 2000C has two radios, a main U/VHF radio capable of operating on both UHF and VHF frequencies, and an auxiliary UHF-only radio.Navigation systems: The Mirage 2000C has an onboard inertial navigation system (INS), which uses three gyroscopes to measure acceleration in each axis. This is integrated over time to track changes in position. It must be initialized with the aircraft’s starting position on the ground, and can be corrected in-flight against known landmarks. In addition, the Mirage has VOR, TACAN, and ILS receivers for radio navigation, and a radio altimeter for measuring height above the ground.IFF system: The Mirage 2000C has an IFF (identify friend/foe) transponder capable of responding to mode-1, mode-2, mode-3, and mode-4 IFF interrogations, and transmitting interrogations to other aircraft.Radar system: The Mirage 2000C has an RDI pulse-doppler radar with multi-mode capability. It’s optimized for air-to-air combat and has very basic air-to-ground capability. It can guide the radar-guided Matra Super 530D missiles that the Mirage can carry.Radar warning system: The Mirage 2000C has a radar warning receiver (RWR) that detects incoming radar signals, analyzes their source, and provides warnings to the pilot. In addition, the Mirage has a missile warning system (MWS), a series of cameras mounted around the aircraft that detect incoming missiles by their distinctive smoke plumes.Weapons systems: The Mirage 2000C has four wing pylons, four lateral fuselage pylons, and one center fuselage pylons for a total of 9 hardpoints to carry weapons. The Mirage can carry heat-seeking and radar guided air-to-air missiles, unguided rockets, bombs of various types, and external fuel tanks.Countermeasures system: The Mirage 2000C has a built in radar jammer (“SABRE”) that can jam enemy radars. It also has internal chaff and flare cartridges that can launch chaff (spoofs enemy radar and radar-guided missiles) and flares (spoofs enemy heat-seeking missiles). The Mirage can also carry an Éclair pod that carries additional chaff and flare. The Eclair pod can be mounted where the arrestor hook or drag chute would otherwise go.Okay, that should be about all you need to know to understand the various switches, knobs, dials, and gauges in the cockpit. So without further ado, let's get to it!As before, we'll start from the left side and work our way around the cockpit to the right side. Also, as with the MiG-21, the labels will be in English but are of course in French in most actual cockpits.The left side has the fly-by-wire controls, audio controls, and radar controls.The back-most panel on the left side is the fly-by-wire test system. The “FCS 5” switch activates the emergency fifth fly-by-wire channel. This provides degraded FBW operation in case the other four channels fail.The two “TEST” switches test the autopilot (“AP”) and flight control system (“FCS”) computers. When the test completes, a green or red light will illuminate indicating the outcome of the test.Moving forward, we have four guarded switches. The furthest switch labeled “A/B OFF” is the emergency afterburner cutoff switch, used to terminate the afterburner if the thrust control cannot do so.The “EMER OIL” switch enables the emergency oil pump, providing oil pressure to the engine in the event the engine-driven oil pump fails.The “ENG COMP” switch controls an emergency secondary engine computer. It’s used when the main engine computer fails. It’s a three-position switch that enables the secondary computer, disables it, or forces a reset of the main engine computer.The “FUEL DUMP” switch opens the fuel dump valve, which dumps fuel in the external tanks only. Once opened, it cannot be closed.Forward of the four emergency switches are the trim controls. The “TRIM DIRECTION” rocker trims the rudder left or right. The large wheel next to it switches between normal and emergency trim modes.Further from that panel are two switches against the sidewall and a third recessed switch. The “RADAR GROUND TRANSMISSION” switch, when on, allows the radar to function while the aircraft is on the ground (normally this is disallowed for ground crew safety).The switch forward of that activates the tape recorder, which records the HUD and cockpit view.The recessed switch is the air relight switch. This switch enables the mid-air relight process, which is used to restart a stalled engine in midair.Ahead of the relight switch is the audio control panel. It consists of a series of volume knobs for the various audio systems in the aircraft. The large “AMPLIS” switch is the amplifier selector switch. This switch chooses between one of two amplifiers for the headset, in case one fails.On the seat, there is an oxygen “TEST” button, an “EMER RELEASE” lever that turns on the valve to the emergency oxygen supply, a “TEST SURP” lever that tests the oxygen mask, and a “100%” lever that toggles between 100% oxygen and an altitude-dependent mixture of oxygen and air.Moving forward, we have the radar control panel adjacent to the thrust lever.The recessed switch above the thrust lever controls the air refueling valve and lights. It has three positions: off (no refueling), day (refueling), and night (refueling plus lights).At the back row, the “RANGE” rocker increases or decreases the radar display range. The “STT” button moves the radar to single-target-track mode, where it focuses all its energy on tracking a single target (typically to support a radar-guided missile launch). The “SCAN” switch chooses the azimuth (width) of the radar sweep — from 60° to 15° of the sky laterally.The “STORE” switch is used to choose how long radar returns stay on the scope before they fade out. The “HFR”/“BFR” switch selects the pulse repetition frequency (PRF) — HFR (haute fréquence; high frequency), BFR (basse fréquence; low frequency), or ENT (entrelacé; interleaved). The high PRF mode gives you longer radar detection ranges and better “look-down” performance (detecting targets below you). Low PRF mode gives you better performance detecting slower targets. The ENT mode interleaves (cycles) between the two modes. The “BARS” switch chooses how wide the radar scans vertically — 1, 2, or 4 bars. Each “bar” is a left-to-right or right-to-left sweep of the radar at a different pitch angle.Forward of those is the “PRI”/“B” switch, which toggles the radar scope between PRI mode and B-scope mode. In PRI mode, the radar display is rendered like a cone coming out of the aircraft’s nose, and targets appear relative to their actual position in space. In B-scope mode, the bottom of the cone is stretched out to form a square, and targets closer to the aircraft are more stretched out laterally than those further away.The “A” button next has an unknown function. The switch next to that controls the target designator cursor (TDC) mode. The TDC is the “mouse cursor” on the radar display used to designate and lock targets. In “ANG” mode, the number next to the cursor shows the altitude of the center of the radar cone at the distance of the TDC from the aircraft. In “ALT” mode, the numbers next to the cursor show the minimum and maximum altitudes of the radar cone at the distance of the TDC from the aircraft. These numbers change as the pilot slews the TDC up and down the radar display.The “TEST” button is also used by maintenance personnel. The “CUT” button activates the radar’s ground avoidance mode, where the radar is used to map the terrain ahead of the aircraft for ground avoidance. The “TER” button activates the ground-mapping mode, displaying the terrain ahead on the radar scope.The large knob controls the radar power mode. It has the positions OFF, WARM UP (préchauffage; power on and warming up), SIL (silence; power on but not transmitting), and EM (emission; power on and transmitting).Above that knob, the RESET button performs a reset of the radar. Next to that switch is the Doppler reject switch, with positions “WITH” (avec), “AUTO” (aut), and “W/O” (sans). Doppler reject is a function that automatically hides from the radar display contacts that are moving so slowly that they’re probably either ground vehicles or the ground itself. In “AUTO”, the Doppler notch (the speed range that is filtered out) is automatically controlled. The other two positions increase or decrease this range manually.The “GAIN” knob controls radar gain, which is the intensity that radar returns must be to be separated from background noise.The unlabeled forward two knobs are separated by a “VAL” button. The further knob is non-functional, and the closer knob is a signal filter control. The function of the button is unknown.The yellow-jacket handle, when pulled, activates the emergency secondary thrust lever. This small lever underneath the handle controls the engine in emergency fuel mode.The large handle at the top of the sidewall (just cut off in the screenshot above but visible in others) controls the drag chute, which is deployed to slow the aircraft down after landing when necessary. On some aircraft, an emergency tail hook is installed instead, and this handle controls the tail hook. The tail hook catches an arrestor wire on the airbase, used to stop the airplane quickly in the event of a brake failure.Moving forward, we have the flight control systems. The “SCOOPS” switch toggles the engine intake scoops between AUTO (automatically controlled) and R (retracted). The “S/CONES” switch toggles the shock cone between AUTO and R (retracted). And the “SLATS” switch toggles the slats between AUTO mode, and manually extending or retracting them.Forward of that panel is the exterior lights panel. The “ANTI COLL” switch turns the anti-collision lights off, or turns them on in low- or high-intensity mode. The anti-collision lights are bright flashing white lights.The “NAV LIGHTS” switch controls the navigation lights, in off, low-, or high-intensity mode. The navigation lights are red, green, and white lights around the aircraft.The “FORM LIGHTS” switch can also be set to off, low, or high. The formation lights are yellow lighted strips around the aircraft used to visualize formation position at night.The switch next to that panel against the sidewall activates the SERPAM flight data recorder or “back box”.Moving forward, we have controls for the two radios. But first, we’ll do the two additional switches up against the sidewall. The forward-most switch controls the police light, which is used to illuminate an unknown aircraft at night to help identify it. The wider rear switch activates the landing or taxi lights, used when landing or taxiing at night.OK, so the top radio set is the secondary UHF radio (“red” radio). The top-left switch is “5W”/“25W” selects between amplifiers and is always set to 5W. Next to it, the “MUTE” switch turns on or off squelch, which automatically cuts out static. The “F+A2” switch is a test selector used by maintenance personnel.The “CDE” button activates encrypted audio; it turns green when audio is encrypted. The knob next to that button controls the radio mode: OFF, ON, F1 (fixed preset frequency; not sure about this one), and H (homing, used to navigate to the source of a radio transmission). The “TEST” button tests the radio.The large knob selects the radio frequency. Frequencies are programmed into one of 20 presets. The window to the right of the knob shows which preset is active. There is no way to manually tune a radio frequency for the secondary radio.Below the secondary radio controls are the primary (“green”) radio controls. At the top left is another “TEST” button. The “E+A2” switch activates the encrypted mode. The “MUTE” switch toggles squelch on and off.Below those switches are windows where the pilot can manually set the radio frequency (there’s an up/down rocker next to each digit). The large knob, as with the secondary radio, selects which of 20 preset frequencies to use.Below the frequency window is the main mode switch. The modes are OFF, PRI (transmits/receives on the selected frequency), PRI+G (same as PRI but also receives on the “guard” frequency, 243.0 MHz, which is a worldwide emergency frequency), F1 (same as with the secondary radio), and H (homing, same as with the secondary radio).As with the secondary radio, the “5W”/“25W” switch selects between amplifiers and is always set to 5W.The frequency mode dial to the right of that switch selects between M (uses the manually-dialed frequency), P (uses the preset frequency), and G (uses guard frequency).The large yellow-jacket paddle is the emergency canopy jettison handle.Moving up to the vertical panel, we have the large red landing gear lever, which raises and lowers the landing gear. Above it is the large yellow-jacket emergency landing gear release handle, which when pulled, lowers the landing gear in the event of a hydraulic failure. The large yellow-jacket button is the emergency jettison button, which jettisons all stores on the aircraft.Below that button are a series of lights showing the status of the landing gear (three green lights means all three gear are down and locked), the air brakes, the parking brake, drag chute or arrestor hook, and nose wheel steering system.The “FCS GAIN” switch activates the emergency fly-by-wire gains. This is used when the normal fly-by-wire computer has a malfunction. Emergency gains is activated for the rest of the flight.The switch above the FCS GAIN switch arms the guns. The switch to the right of the FCS GAINS switch toggles between air-to-air and air-to-ground g limits, allowing the aircraft to pull more g when no bombs are loaded.Below those switches are a row of “level lines” indicating the control deflections that the FCS is sending to the four elevons and the rudder.The small box attached to the edge of the left eyebrow controls the countermeasures system. The “AUTO”/“MAN”/“OFF” switch activates the chaff and flare dispensers, and turns on or off automatic countermeasures dispensing based on a detected threat on the RWR. The two numeric displays show number of chaff and flares remaining in the pod. The “PRGRM”/“S REL” switch toggles between releasing the selected chaff/flare program or launching a single chaff and flare with each release.Next to the countermeasures box is a clock with winding and setting capability.On to the left forward console. Angled along the eyebrow sill are the autopilot controls. The “TEST” button tests the autopilot lights. Each of the other buttons activates an autopilot mode, which lights up the green light when active, or the yellow light when armed. From left to right, you have the autopilot master (activates attitude hold), current altitude hold, selected altitude hold, an unlabeled button (which in some aircraft activates a ground-controlled intercept mode), and an approach mode (that follows an ILS signal to a runway).The tall vertical barber pole indicator is angle of attack, with the green range being the acceptable landing angle of attack. To the left are the yellow master caution and red master warning lights; pressing the light acknowledges the alert. Below that are the radio repeaters, that display the currently-tuned frequency (preset or manual) for the UHF and U/VHF radios.The two blue-and-black balls are attitude indicators (primary and backup) that display the aircraft’s orientation against a virtual horizon. The larger primary attitude indicator also displays current heading, making it a “navball”. The primary attitude indicator has a cage lever to cage the gyro when not in flight, and the backup attitude indicator has an index knob used to raise or lower the symbolic “wings”, as well as cage the gyro.The “NORM”/“SPIN” switch, when set to SPIN, temporarily disables the fly-by-wire limits on angle of attack, allowing the pilot to exceed maximum angle of attack. This is used to recover from a spin (vrille).Below that, the five-digit window is used to select the altitude for the autopilot to hold. To the right of that is the vertical speed indicator, indicating rate of climb or descent in feet per minute. Below and to the left of that is the airspeed indicator (with a window for Mach speed), then the altimeter (with digital display and a knob for setting current sea level pressure).Below that is the PCA (poste de commande armament or armament control panel). It consists of a red master arm switch, a guarded yellow-jacket selective jettison switch, and a series of buttons whose function changes depending on the current weapons mode. Currently displayed button actions are “TOP” (activates a time-over-steerpoint mode that assists you in getting to a location at a certain time), “POL” (police mode; activates the radar for locating targets but does not allow weapons firing), “APP” (activates approach mode, assisting the pilot in flying an approach to a runway), “RD” (route désirée; activates normal flight plan following mode), and “OBL” (recalage oblique de la centrale; activates a mode that recalibrate the INS using the radar).The bottom row has a button for each weapon loaded onto the aircraft. “MAG” refers to the Magic II air-to-air missiles, “530” to the Matra Super 530D radar-guided missiles, and “RP” to the external fuel tanks (réservoirs pendulaires). You can press a button to select a weapon to fire (or jettison, if the selective jettison switch is active).Moving to the HUD control panel just below the HUD, we have at the top-left the DCLT (declutter) switch, which when active removes some symbology from the HUD to clear it up. The “CLEAR” button temporarily declutters the HUD when held.The “WGS DEP” knob sets the manual target wingspan in meters. This is used on the HUD to display a guide for aiming the gun. The pilot uses the HUD to line up the target’s wings with an aiming guide used to estimate distance.The “CCTL” switch toggles the gunsight between CCLT mode (calcul continu de la ligne de traceurs; continuous calculation of the path of the bullets) and PRED (prédéfeni; uses a predefined target distance).The “BRT” knob on the left controls HUD brightness, and the “STBY BRT” knob on the right (mislabeled) sets the position of a backup, fixed gun cross. As a backup, the pilot can manually calculate the deflection necessary for a shot, and dial it into the backup gun cross.The “SELH” switch controls which altitude is shown on the HUD. The “ZP” position shows barometric altitude and the “H” position shows radar altimeter height.The “RAD ALT” switch is used to turn ON or OFF the radar altimeter, and put it into self-test mode. The knob next to the switch sets the minimum altitude. If the aircraft descends below that altitude (on the radar altimeter), the pilot will be warned.There are two unlabeled switches below the HUD control panel. The left one turns on and off the HUD, and the right one toggles on or off the backup fixed sight.Below that is the radar display. Along the sides of the display are eight unlabeled switches that control the radar’s display of ground-controlled intercept targets. These targets are transmitted to the aircraft from ground controllers and displayed on the radar scope. On the left side, the switches are (from top to bottom): start/finish input designation (unsure about this one), number of objectives (also unsure), display polar coordinates, and display bearing coordinates.Along the right side, the witches are: display course to GCI target, display GCI target altitude, display GCI target mach number, and display time that GCI target was relayed to the aircraft.Moving below the radar display to the center pedestal, the “DECLT” switch declutters the radar display. The “TV/RDR” switch (mislabeled) controls the location of the ground radar map relative to the symbology; the pilot can shift it up or down to line it up with the symbology.The four numeric thumbwheels control (from left to right) marker brightness (unsure about this), backlight, radar contrast, and radar brightness. The “TDF”/“TDR” switch turns on or off the display.Below that is the IFF controls. There are two sets of thumbwheels for setting the mode-1 and mode-3 IFF codes. Mode-1 is used to set the mission number, and mode-3 is used to set a code given to the aircraft by air traffic control for identification purposes. Mode-2 is an aircraft identification number and it’s hard-coded.Below that are switches that activate each of the IFF modes: mode-1, mode-2, mode-3A (ATC identification), and mode-C (altitude reporting to ATC). The “IDENT” switch, when set to IDENT, sends an ATC identification signal that helps radar controllers locate the aircraft on their scope (when requested). When set to “MIC” the identification function is performed every time the push-to-talk button on the thrust lever is pressed.Below and to the left of those switches are the IFF mode-4 controls. Mode-4 is the only encrypted mode and therefore the only true identify friend/foe mode. The mode-4 IFF can store two encrypted IFF codes (labeled “A” and “B”). The mode knob selects between codes A and B. In the “ZERO” position, the two codes are erased. Normally they are also erased after the aircraft is turned off, unless the knob is set to “HOLD”.The ON/OFF switch turns on or off mode-4 replies. The light illuminates when replying to a mode-4 interrogation. The “AUDIO”/“OFF”/“LIGHT” switch controls what happens when replying to a mode-4 interrogation: an audio alert can be played, or the light illuminated.To the right of the mode-4 controls are the master IFF controls. The large knob selects the main mode: off, SBY (standby), N (normal operating mode), or EMER (replies to mode-3 ATC interrogations with a code indicating aircraft in distress). There is also a TEST button and a light indicating an IFF fault.At the bottom of the center pedestal are two gauges. The left gauge indicates pressure in the two hydraulic systems, and the right gauge indicates cockpit pressure altitude (the air pressure in the cockpit, shown as an altitude).The yellow-jacket handle is the eject lever.Moving to the right forward panel, the topmost gauge is the accelerometer, showing g forces on the aircraft.Below that is the radar warning receiver, which indicates the location of detected radars scanning this aircraft. The row of LEDs below the RWR indicate the status of the countermeasures system: S (standby), ECM (jamming), RWR (RWR is on), MWS (MWS on), and J (chaff or flares dispensing). The knob controls RWR brightness.Below the RWR is the horizontal situation indicator (IDN, indicateur de navigation). It indicates the aircraft’s heading. The blue index shows the commanded heading for the autopilot. There two arrows indicate the bearing to the current steerpoint or the currently tuned TACAN station. The inset numeric readout is the distance to the current steerpoint or TACAN station. Along the inside of the bottom is the current navigation mode: TH/NAV (true heading and navigating INS steerepoint), NAV (navigating to INS steerpoint), TAC (navigating to TACAN station), CFS (navigates to an offset position from the INS steer point), [math]\rho[/math] (unlabeled; sets offset distance), [math]\theta[/math] (unlabeled; sets offset bearing), and DTL (guided by remote GCI datalink).The left +/- knob below the IDN sets the offset distance or bearing when in [math]\rho[/math] or [math]\theta[/math] modes. The right knob changes the navigation mode.Moving back up to the eyebrow, the “A/B” light illuminates when in afterburner. Below that is the “N” gauge, showing engine RPM as a percentage of max RPM. Below that is the “T7” gauge, showing exhaust gas temperature in hundreds of °C.Below the engine gauges is the combined fuel gauge. At the top are two digital readouts showing the total amount of internal fuel (left) and internal + external fuel (right). These fuel amounts are not directly sensed; they are calculated by a totalizer based on fuel flow. The “PRESET” switch increases or decreases the total fuel amount manually, and should be used when refueling.The orange light at the top activates when air refueling is ongoing.The left vertical pole gauge indicates directly-sensed internal fuel in the left feeder tank, and the right pole gauge in the right feeder tank. The feeder tanks feed directly to the engine, so they empty last.The matrix of lights indicates which fuel tanks are empty. There is a light for each tank.The “TRANSFER” switch tests the fuel transfer circuit. The “CROSSFEED” knob, when turned, opens the tank gates, allowing fuel to flow freely between the left and right tank groups (for balance purposes).To the left of the fuel gauges is the weapons configuration panel (PPA, poste de préparation armement). The “L”/“R”/“AUTO” switch selects which missile (Magic or 530D) will be launched, the left or right. In AUTO, the missile closest to the target’s position off the nose will be launched.The “P MIS” button powers on the Super 530D missile “P” displays when the missile is ready to be fired. The button to its right is not used. The “P MAG” button powers on the Magic II missiles, and “P” displays when ready to fire. The “TEST”/“PRES”, when in TEST, tests all lights on the PPA. When in “PRES”, displays a pictorial representation of the weapons on the aircraft on the radar display.On the bottom row, the leftmost switch sets bomb fusing: INST (bombs explode instantly on impact), RET (retarded, bombs are allowed to penetrate into the target briefly before exploding), and INERT (bombs do not explode).The “QTY” switch sets the number of bombs to be released in one salvo. The “DIST” switch sets the distance between impacts for the salvo. The values for each of these is read off the LED screens to the right of the switches. The “CAN ROF” button sets firing mode for the guns, rockets, and Super 530D missiles. In PAR, a single 530D missile is launched, or sets burst mode for rockets and guns. In TOT, both missiles are launched, or sets continuous fire mode for rockets and guns.Alright, almost there! Time to do the right side panels.Here we’ve got the master electrical switches, warning lights, navigation controls, countermeasures controls, engine starting, and interior lighting controls. We’ll start forward and work our way back.The big red handle opens, closes, and locks the canopy. Below it is a round wheel just barely jutting out from under the canopy sill; that’s the emergency compass. It can be pulled out from its stowed position to show magnetic heading when the INS fails.Going to the vertical panel, at the very top, we’ve got a liquid oxygen (LOX) quantity gauge.Below that we’ve got the main red “BATT” switch, which turns on and off the battery. To the right of that are on/off switches for the transformer-rectifier, and the two alternators. To the very right is a lights test switch, which tests the matrix of warning lamps below.To the right of the warning lamps are two switches. The top switch I’m not sure about, but the bottom switch holds the aircraft into an “alert” state, where it is mostly ready for takeoff, and can be scrambled at a moment’s notice.The forward part of the horizontal sidewall is the navigation control panel or PCN (poste de commande navigation). The control panel is used to enter the latitude and longitude of waypoint, navigate to those waypoints, set target data, and do other navigation functions.The large knob sets the INS parameter, which is the field that is currently being displayed or edited. These parameters can apply to the aircraft’s current position or to a steerpoint. Each knob position represents a pair of numbers. There are two LED windows at the top of the control panel, and the two numbers go into the two LED windowsOptions are RD/TD (route désirée/temps désiré; desired bearing and time), L/G (latitude and longitude), ALT (altitude [in feet and meters]), RH/DS (runway heading and glideslope for approach).In addition, there are also display-only modes: DT/DTM (distance and bearing to next steerpoint), TR/VS (time to next steerpoint and ground speed), GS/ΔTM-So (mislabeled, wind direction and speed), F PLN CT/WS-WD (mislabeled, magnetic declination).And in addition, there are three editable offset modes, for setting an offset waypoint from a waypoint: [math]\rho[/math]/[math]\theta[/math] (distance and bearing offset from waypoint), ΔALT (altitude offset from waypoint), and ΔL/ΔG (latitude and longitude offset from waypoint).Below the two data fields are two two-digit numeric fields. The left contains the number of the waypoint currently being edited, and the right the number of the waypoint currently being navigated to (the steerpoint).Pressing the “PREP” button below the left waypoint field lets the pilot choose the waypoint to be modified using the numeric keypad. Pressing the “DEST” button below the right waypoint field lets the pilot choose the steerpoint.The “ENC” (enchaînement) button between the two turns on auto-navigation, which automatically sequences the steerpoint to the next waypoint once the current steerpoint is passed.To the right of the mode switch are three more buttons. The top button, BAD (but additionnel), activates the offset waypoint that was programmed in using the offset parameters. The middle button, REC (recalage) starts an INS position update process, allowing the pilot to overfly a waypoint at a known visual landmark to re-align the INS. The bottom MRQ (marquage) button marks the current position as a markpoint, storing it for later navigational use.The button just below the mode switch labeled VAL (validation) is pressed to begin INS alignment, accept in-flight INS position updates, and accept markpoint data.The keypad is used to enter numeric values for parameters, and it also has EFF (effacement; clear) and INS (insertion; enter) buttons.Moving down, we have the IFF interrogation panel. The left knob selects which kind of IFF interrogations will be sent: mode-1, mode-4, modes 2 and 3, mode-3, or modes 3 and 4. The four thumbwheel digits set the mode-3 code to interrogate. Any other aircraft with this mode-3 code set will come back as friendly for a mode-3 interrogation. The switch to the right of the thumbwheel turns on and off power to the interrogator.Below the IFF controls are the ECM controls. The leftmost switch sets the ECM mode: STBY (standby), a square (normal mode), or PCM (priorité contremesure; jammer takes priority over radar, causing you to jam your own radar as well).The three switches to the right have ON, OFF, and TEST positions: “ECM” (controls the jammer), “RWR”, and “MWS”.The sub panel to the right controls the countermeasures program. The switch sets the dispenser mode: AUTO (chaff and flare dispensed automatically when a threat appears on the RWR), SA (semi-auto; chaff and flare are dispensed when the pilot commands in), and OFF (no chaff and flare dispensed).The large knob selects among different predetermined chaff and flare dispense programs. There are 10 such programs, and each dispenses chaff and flare in different quantities and with different timings designed to fool specific missiles or radars. When the knob is in “A”, the program is automatically selected depending on the primary threat the RWR detects.The bottom-most panel before the gap controls the VOR/ILS and the TACAN. The VOR/ILS subpanel allows the pilot to tune a VOR station that s/he can navigate to, or an ILS frequency to use to approach a runway. The two knobs tune the major and minor parts of the frequency. Around the outer ring of each knob is another selector knob. The left outer knob turns on and off the receiver, and the right outer knob selects between the two receiver antennas: HG (haut-gauche; high left) or BD (bas-droit; low right).The TACAN subpanel allows the pilot to tune a TACAN station. TACAN frequencies are referenced by channel number (1 to 126) split into two bands (X and Y). The left and right knob set the digits of the TACAN channel. The outer ring of the left knob toggles between the X and Y bands, and the outer ring of the right knob sets the TACAN mode: OFF, REC (receive only), T/R (transmit/receive), and A/A (air-to-air). In receive-only mode, the TACAN only receives bearing information. In transmit/receive mode, the TACAN also transmits radio pulses to the station, and measures the time of the reply to calculate distance to the station. Air-to-air mode is used when navigating to an airborne TACAN, which are placed on air refueling tankers.Home stretch! The forward-most panel below the gap sets the INS mode. Options are: OFF, STBY (powered but not operating), CAL (maintenance calibration mode), TST (test mode), ALN (starts aligning the gyroscopes, an 8-minute process that’s necessary before the INS can be used for navigation), STH (stored heading, a faster alignment using stored information from the last time the aircraft was shut down, assuming it hasn’t been moved since then), NAV (normal navigation mode), and EMG (emergency mode, when alignment data is lost, allows basic attitude navigation).To the right of the INS mode is a port to insert a data cartridge containing INS waypoint data and loadout data. To the right of that is the INS operational mode selector: a boxed N (normal mode), STS (status mode, shows the INS alignment progress), CDI (données codées inertielles; displays coded inertial data for maintenance), FFR (C/R de vol; displays a maintenance report), and MTN (maintenance mode).Below the INS controls are the environmental controls. We’ve got two big orange buttons labeled “H” and “C”, these set the avionics cooling mode to hot or cold.To the left, the “AUTO”/“MAN” switch toggles the air conditioning between automatic or manual mode.To the right, the “COND” switch turns on and off the air conditioner, and below that, the “DEPOLL” switch (unsure about that).The large knob controls cockpit temperature. The top half sets the set point for automatic temperature control, and the bottom half is used for manual temperature control.To the right, the “DEMIST” switch toggles canopy defogging.Below the ECS controls are the interior lighting controls. The “UV” knob has an inner and outer ring. The outer ring controls the brightness of the red flood lights, and the inner ring controls the brightness of the instrument panel backlights.The “CONSOLE” knob also has two rings; the outer ring controls the brightness of the lights for the left and right side consoles, and the inner knob for the front panels.The “NIGHT”/“DAY” switch toggles the brightness of the caution and warning lights, and the “WHITE” knob controls the brightness of the white flood lights.Behind another gap is the engine start controls. The big guarded red button is the starter switch, used to start the engine. Opening the guard also reveals the on/off switch for the starter fuel pump, which is used to provide fuel to the engine during start.The “L/H” switch controls which ignition system is used for engine start, left or right. There is also a position called VENT that motors the engine without fuel or ignition; it’s used to clear gases from the engine after a bad start.The “PUMPS” switches turn on the left and right fuel pumps, which pump fuel from the left and right tank groups. The smaller guarded switch labeled “FP MAN COCK” is the fuel shutoff valve, which when closed, shuts off all fuel to the engine.Behind the starter controls are circuit breakers. The large handle is the parking brake. The switch on the seat adjusts seat height, and the yellow handle is used to manually separate the seat from the pilot after an ejection (should automatic separation not occur).Alright, that’s about everything! Thanks for joining me on this aventure incroyable!

What is Mathematics? Aristotle defined mathematics as “The science of quantity”, while Isidore Auguste Comte preferred calling it “the science of indirect measurement” and Benjamin Peirce “the science that draws necessary conclusions”.The answer changes depending on the philosophical stance of the definer, and on the branch of mathematics s/he wishes to focus on. And, as new branches of mathematics are discovered and developed, the definition also continues to develop, adapt and change accordingly.Scientists have long used mathematics to describe the physical properties of the universe. But what if the universe itself is math? That’s what cosmologist Max Tegmark believes.In Tegmark’s view, everything in the universe — humans included — is part of a mathematical structure. All matter is made up of particles, which have properties such as charge and spin, but these properties are purely mathematical, he says. And space itself has properties such as dimensions, but is still ultimately a mathematical structure.Mathematical Structure is involved in Everywhere“If you accept the idea that both space itself, and all the stuff in space, have no properties at all except mathematical properties,” then the idea that everything is mathematical “starts to sound a little bit less insane,” Tegmark said in a talk given Jan. 15 here at The Bell House. The talk was based on his book “Our Mathematical Universe: My Quest for the Ultimate Nature of Reality” (Knopf, 2014).“If my idea is wrong, physics is ultimately doomed,” Tegmark said. But if the universe really is mathematics, he added, “There’s nothing we can’t, in principle, understand.”The idea follows the observation that nature is full of patterns, such as the Fibonacci sequence, a series of numbers in which each number is the sum of the previous two numbers. The flowering of an artichoke follows this sequence, for example, with the distance between each petal and the next matching the ratio of the numbers in the sequence.The branches and leaves on a sneeze wort plant grow in a Fibonacci patternFibonacci spiral in hurricaneTalking about Fibonacci Series or spiral another thing is associated with it is Golden Ratio which is the most elegant irrational number after π and e denoted as φ , the limit of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence), as originally shown by Kepler.This is found in all most every where in the worldThe seed heads are so tightly packed that total number can get quite high — as many as 144 or more. And when counting these spirals, the total tends to match a Fibonacci number. Interestingly, a highly irrational number is required to optimize filling (namely one that will not be well represented by a fraction). Phi fits the bill rather nicely.In the Vitruvian Man, when vertical lines are drawn from the wrist to the elbow and from the fingertip to the wrist, the ratio of these proportions is 1:1.61803. This particular ratio is called the Golden Ratio. This ratio is replicated in all other areas of the painting. The most basic geometric construction of the Vitruvian man shown above is the same for every human bodyFor a perfect smile, the front two teeth form a golden rectangle. There is also a golden ratio in the height to width of the center two teeth. And the ratio of the width of the two center teeth to those next to them is phi. The ratio of the width of the smile to the third tooth from the center is also phi.From the illustration given below, we can see several occurrences of the golden ratio found in thehuman body. 1. Sole to navel: Sole to crown. 2. Sole to knee: Sole to navel. 3. Navel to shoulder: Navel to crown. 4. Knee to calf-muscle: Knee to sole. 5. Navel to mid-thigh: Navel to knee. 6. Navel to mid-chest: Navel to base of throat. 7. Base of throat to temple: Base of throat to crown. 8. Calf muscle to ankle: Calf muscle to sole. 9. Mid-thigh to start of kneecap: Mid-thigh to end of kneecap. 10. Navel to crotch: Navel to mid-thigh. 11. Navel to sternum base: Navel to sternum or mid-chest. 12. Base of throat to earlobe: Base of throat to top of ear. 13. Brow bone to hairline: Brow bone to crown. 14. Nose to chin: Nose to base of throatIn the human lungs, the windpipe divides into two main bronchi, one long (the left) and the other short (the right). This asymmetrical division continues into the subsequent subdivisions of the bronchi. It was determined that in all these divisions the proportion of the short bronchus to the long was always 1/1.618.Even the DNA molecule, the program for all life, is based on the Golden section. It measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral.34 and 21, of course, are numbers in the Fibonacci series and their ratio, 1.6190476 closely approximates Phi, 1.6180339.Mekke City's proportions in distance to the north and south poles and also the proportion of the eastern & western elongation are equal to Golden Ratio.the latitude and longitude position of the Holly Kaaba are lat: 21.42251 and lon: 39.8262 or 21o25’21.04”N and 39o49’34.32”E. The position of the Makkah city centre, pointed on Google Earth, is lat: 21.42737 and lon: 39.81483 or 21o25’38.55”N and 39o48’53.41”E. For both of these positions, the ratio between the south and the north parts is almost 1.624 or 1.6 (approx.) and the ratio between the distance to the west and the east is approximately 1.568 or 1.6 (approx). Both of these ratios are closer to the value of the Golden Ratio, 1.61803. The distance between the Golden Mean point of the Earth and the Holly Kaaba is just a few kilometres, 277km, and if a straight line is drawn between these two points, shown , Mena falls just on it.Dimension of Kabba also maintain Golden RatioHere, we have repetetive nummerical values from the table below for example chapters 85 and 99 have same nummerical value 107, we summerize all repetitive nummerical valuesHere, we have non-repetitive nummerical values, and we summerize them alsoWe can clearly see golden ratio between Reptitive and non-repetitive nummerical values from this tableThe nonliving world also behaves in a mathematical way. If you throw a baseball in the air, it follows a roughly parabolic trajectory.Parabolic arch concrete bridge carrying alresford road over winchester-bypassSydney Harbour BridgePlanets and other astrophysical bodies follow elliptical orbits.“There’s an elegant simplicity and beauty in nature revealed by mathematical patterns and shapes, which our minds have been able to figure out,” said Tegmark, who loves math so much he has framed pictures of famous equations in his living room.One consequence of the mathematical nature of the universe is that scientists could in theory predict every observation or measurement in physics. Tegmark pointed out that Mathematics predicted the existence of the planet Neptune, radio wavesand the Higgs boson particle thought to explain how other particles get their mass.Some people argue that math is just a tool invented by scientists to explain the natural world. But Tegmark contends the mathematical structure found in the natural world shows that math exists in reality, not just in the human mind.And speaking of the human mind, could we use math to explain the brain?Some have described the human brain as the most complex structure in the universe. Indeed, the human mind has made possible all of the great leaps in understanding our world.Someday, Tegmark said, scientists will probably be able to describe even consciousness using math. (Carl Sagan is quoted as having said, “the brain is a very big place, in a very small space.”)“Consciousness is probably the way information feels when it’s being processed in certain, very complicated ways,” Tegmark said. He pointed out that many great breakthroughs in physics have involved unifying two things once thought to be separate: energy and matter, space and time, electricity and magnetism. He said he suspects the mind, which is the feeling of a conscious self, will ultimately be unified with the body, which is a collection of moving particles.But if the brain is just math, does that mean free will doesn’t exist, because the movements of particles could be calculated using equations? Not necessarily, he said.One way to think of it is, if a computer tried to simulate what a person will do, the computation would take at least the same amount of time as performing the action. So some people have suggested defining free will as an inability to predict what one is going to do before the event occurs.But that doesn’t mean humans are powerless. Tegmark concluded his talk with a call to action: “Humans have the power not only to understand our world, but to shape and improve it.”So,Mathematics is profoundly effective at describing the world around us.It is the language physicists use to formulate theories about our universe,Newton's Second Law of motion,1687de Broglie Matter waveMax Planck's Quantum TheoryNewton's Law of GravitationMaxwell's Equation of ElectrodynamicsEinstein's Mass Energy EquivalenceEinstein's Field Equation of General RelativitySchrödinger equation of Quantum MechanicsNeurologists use to model our brain,Mathematical modeling of memoryIn Medical Science ,Doctors also need mathematics for Medicine Quantity like action and decay of drug, cardiac issue, hormonal behavior etc.(Mathematical Modeling in Biological Science Cardiac output is the rate R of volume of blood pumped by the heart per unit time (in liters per minute). Doctors measure R by injecting A mg of dye into a vein leading into the heart at t = 0 and recording the concentration c(t) of dye (in milligrams per liter) pumped out at short regular time intervals (see figure). Assume A = 7 mg. Estimate R using the following values of c(t) recorded at 1-second intervals from t = 0 to t = 10. (Round your answer to two decimal places.) t (sec) 0 1 2 3 4 5 6 7 8 9 10 c(t) 0 .5 2.9 6.6 9.9 9.2 6.3 4.2 2.5 1.2 0 = Liters/min)Coordinate Geometry (Graph) is used in Cardiology (ECG).Hardy–Weinberg proportions for two alleles: the horizontal axis shows the two allele frequencies p and q and the vertical axis shows the expected genotype frequencies. Each line shows one of the three possible genotypes.Hardy–Weinberg equilibrium, model, theorem, or law, states that allele and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences. These influences include mate choice, mutation, selection, genetic drift, gene flow and meiotic drive. Because one or more of these influences are typically present in real populations, the Hardy–Weinberg principle describes an ideal condition against which the effects of these influences can be analyzed.(Mathematical Modeling of Female Hormone.Mathematicians have determined that women, when asked how many men they’ve slept with, tend to enumerate, while men approximate. And when you enumerate, you underestimate, and when you approximate, you tend to overestimate. Fun fact: 80% of men’s results were divisible by 5 – take that how you will.)(Mathematical Modeling of Male Hormone. Mathematicians looked at how the female body knew how a month has passed between cycles and how these cycles affect their moods. The entire male populace has been criticizing women’s hormones for centuries, so it’s just fair to look at men’s hormones as well. And will you look at that, men’s are much more complicated)and Economists use to model the stock market.(The Second-Order Differential Equations of Dynamic Market Equilibrium.Differential equations can be used to include the dynamic aspects to economics into a mathematical framework which takes into account the volatility present in economics. The propagation of waves across a medium can be adopted in economics in various forms including structural shocks to the economy, diffusion of monetary stimulus throughout various sectors in the economy and the interaction between producer and consumer when considering current prices and their derivatives. Linear as well as Non-Linear Differential equations have their place in economics, but there is still much work to be done)(Black-Scholes Equation.The model of the stock-price in continuous time was represented by a special type of differential equation (so called stochastic differential equation) which allowed for randomness in stock-price)(Gaussian Normal Distribution.A normal distribution is an arrangement of a data set in which most values cluster in the middle of the range and the rest taper off symmetrically toward either extreme. Height is one simple example of something that follows a normal distribution pattern: Most people are of average height, the numbers of people that are taller and shorter than average are fairly equal and a very small (and still roughly equivalent) number of people are either extremely tall or extremely short.This distribution is maintain not only in Economic aspects but also Biological aspect, Random velocity of Gaseous Particles, Electrical Circuit etc)This naturally leads us to ask the question why mathematics is so effective at describing our universe – a question asked many times before by a number of great minds.“Math is the dialect of the universe”…. What does that even mean? There are things we observe in the universe and sometimes we don’t always understand them. Thats why we invented math. Math is the manual of the universe. A guide to help us understand why things happen the way they do. The universe threw countless mysteries at us and we honed those mysteries using numbers. Numbers interpret the way the universe works and translates it so that we can understand the universe. It is the “Google translate” of everything. We didn’t give quantitative values to things that were qualitative. The universe did. We, however, deciphered the code that the universe had set forth and did it without every using a single word. Math is to real life as binary is to computers. It’s the encryption of how things should happen, and it follows that code down to the letter. Thats because the universe IS math. Math teaches us to learn the ways of everything. There is so much underlying wisdom in math. So much latent potential in math.Math is… underrated. From now on, rather than trying to memorize x and y or the quadratic equation, learn that x and y are values that are susceptible to change and that the quadratic equation represents a pattern that YOU can use to your advantage. Math is a tool more valuable than any other. Learn to use it.To understand these peoples fascination with this enigma, we have to understand the origins of mathematical reasoning and their relationship with logic, before we start speculating the relationship between mathematics and nature itself. I would like to note at this point that the (very brief) history of scientists and philosophers I will be providing is tailored to the context of this article, and that I knowingly, and purposefully omit a lot of information central to understanding these characters’ work, and role in history.Mathematical contemplation has its roots in Antiquity. Even though we can be certain humanity has had the ability to count for a lot longer than just a few thousand years – it was only in ancient Greece that philosophers started concerning themselves with mathematics as we would define it today. It began with geometry – the study of shapes, and arithmetic – counting. Important insights into these disciplines had been made by Pythagoras of Samos (ca. 570 BC – ca. 496 BC) and followers of his school of thought (the Pythagoreans). Rational and Irrational numbers were defined, and geometrical relationships were discovered- however more importantly there was the pioneering Pythagorean insistence on mathematical proof – a procedure based entirely on logical reasoning, by which starting from some postulates, the validity of any mathematical proposition could be unambiguously established. For the first time it wasn’t enough to know that something worked – as it had done for the Babylonian mathematicians – the Greeks were beginning to care about why it worked. This set the stage for the next generation of philosophers to build upon what would become the foundations of our Western thinking. Among this next generation was the famous Plato (ca. 428 BC – 347 BC) who is often credited with formerly establishing the discipline of philosophy, by bringing together topics ranging from mathematics, science, and language to ethics, art and religion. Plato’s main contribution to this article is a concept appropriately called Platonism. In its broadest sense, this is a belief in an abstract eternal and immutable reality, independent of the world perceived by our senses. In this reality, perfect mathematical forms reside – such as the perfect square, natural numbers, and all other mathematical objects. This includes all ‘objective truths’ – things which are true, even if we do not know them to be.A Sculpture of Plato made by the sculptor SilanionFermat’s last theorem; a mathematical statement which has been known to be true since it had been conjectured in 1637. It simply says that the equation:can not be satisfied for any three values, provided the value of n is greater than 2. However, finding a proof for this theorem turned out to be an immense challenge, one which was only overcome in 1994 by mathematician Andrew Wiles – over 350 years after it had been formulated! So at which point can it be said that the theorem was a truth? In 1937? In 1994? or was is true all along? The Platonist answer would be that is was true all the time – and furthermore, that this objective truth resides in the realm mentioned above. To the Platonist this reality is as valid as the universe around us. What is important here is that this is the first time the explicit belief is expressed that mathematics is a fundamental ingredient of this universe, which exists outside of the human experience. This is a very profound insight, considering the period of human history this originates from – this is before physics had made so many ground breaking discoveries, based on mathematical modeling and before humans discovered the universal applicability of mathematics. Since antiquity mathematics has been associated with the perfect, divine – but nevertheless as real as anything else in this universe.The great philosopher and mathematician Rene Descartes (1596 – 1650) invented/discovered what we now know as the Cartesian system of coordinates – which intimately linked geometry and algebra to be two sides of the same coin. This allowed mathematicians the algebraically analyze the world around them (which, naturally, is full of geometrical shapes), which lead to immense breakthroughs which would otherwise not have been possible. Even though Isaac Newton despised Descartes (so much, in fact, that he sometimes refused to write his name!), and tried formulating his laws of motion without the Cartesian system – he was not able to, having to eventually concede that the Cartesian coordinate system was the simplest, most logical way to map the physical space. However this was still a time of alchemy and mysticism, and thus the power of mathematics at explaining the world around us was still seen as being inevitably linked with the divine.This ‘divine’ link is part of the reason Euclidean geometry (as derived from Euclid's axioms) was largely unquestioned for more than a thousand years, since it was defined around 300 BC by Euclid of Alexandria. Euclidean geometry is based on 10 postulates (statements which were taken to be indisputably true) from which he sought to prove a large number of geometrical propositions on the basis of logical deductions. The first four postulates were very easy to understand, for instance the first one read:“Between any two points a straight line maybe be drawn”.In contrast, the fifth axiom was considerably less self–evident and slightly more complicated:“If two lines lying in a plane intersect a third line in such a way that the sum of the internal angles on one side is less than the two right angles, the two lines inevitably will intersect each other if extended sufficiently on that side.”Below is Euclid’s 5 postulates are visually represented.A visual representation of Euclid’s postulatesWhilst nobody doubted its validity, it lacked the compelling simplicity of the other axioms. In fact, I am not convinced that Euclid himself was entirely happy with his fifth postulate; in his book The Elements the proofs for his first 28 postulates do not make use of it. Over the years many had tried to deduce an explicit proof for this postulate without any success, forcing the mathematical community to reconsider this divine status given to Euclidean geometry in the case that it did not hold true. In the Nineteenth century the breakthrough finally occurred, and people realized that by choosing an axiom different from Euclid’s fifth resulted in an entirely different, but equally valid geometry. This was huge. For millennium Euclidean geometry was the solid foundation on which all of mathematics and even nature itself was based.The first to publish on the topic of this new, non- Euclidean geometry was Russian mathematician Nikolai Lobachevsky (1792 -1856). In his work, Lobachevsky laid out geometrical relationships on a hyperbolic surface – which simply means that instead of calculating relationships on a flat surface, they were calculated on a curved surface called a hyperbole. Different geometries can be seen in Figure 3 belowHowever, Lobachevsky’s work went largely unnoticed because he published his findings in a rather obscure journal. Independently, Hungarian mathematician Janos Bolyai (1802 -1860) made the same findings.The existence of non – Euclidean geometries had been anticipated, and worked on by the great Carl Friedrich Gauss (1777 – 1855). Gauss is widely regarded as one of the most influential mathematicians of all time, yet he feared that publishing this radically new geometry would be seen as philosophical heresy by Kantian philosophers of his time- who held firm the belief that geometry is somehow linked to the divine. Gauss sent a letter to Janos’ father about his son’s work, expressing his thoughts:This was one of many instances in history, in which a breakthrough in mathematics and/or physics had occurred completely independently at the same time. Another fantastic example is that Calculus was invented/discovered by Isaac Newton (1642 – 1726) at the same time as it was being developed, independently, by Gottfried Leibniz (1646 – 1716) in Germany. It seems as though certain breakthroughs in our understanding of the world seem almost inevitable- leading many to believe that we are indeed discovering mathematics – that mathematics is embedded in nature. However, what the advent on non-Euclidean geometry showed to many, is that axioms could apparently be chosen to give rise to many different mathematical geometries. This no longer made mathematics seem as this divine, perfect model nature was based on. Instead, it had become rather arbitrary, and susceptible to human manipulation. Surely this meant that mathematics was being invented, rather than discovered?For a long time these non–Euclidean geometries were treated as non-physical, amusing curiosities. It wasn’t until Albert Einstein made use of these geometries over one hundred years later, in his theory of general relativity that non–Euclidean geometry was shown to have physical meaning. Besides the obvious fact that non–Euclidean geometry applies on the earths surface (which is round, not flat) it turns out that the fabric of the universe itself has a shape – it is not flat. This is something nobody would have anticipated when Riemann published his works in the 1800s, which would then come in so useful for Einstein about one hundred years later.This brings me onto one of the most curious aspects of mathematics; the fact that quite often theorems or, as Platonist would say, ‘objective truths’ are discovered in mathematics long before they are made use of by scientists non–Euclidean geometry being only one example of a mathematical ‘toy’ turning out to be useful for describing the physical world over one hundred years after its initial publication. Another fascinating example is Knot Theory. This was developed 1771. in order to support a new disproved theory about the structure of matter. According to it, atoms were tightly knotted tubes of ether – a mysterious substance which has now also been shown to not exist. The variety of chemical elements could therefore be accounted for by the variety of different knots in the ether. This sparked a serious interest in classifying knots – which knots are possible, and how many are there. Even though this aforementioned theory of atoms was quickly shown to be false- mathematicians remained interested in knots for no other reason than curiosity. This lead to a very rich understanding of knots, and a lot of conjectures based on them. Imagine, then, the delight when it was found that knot theory is key to understanding fundamental processes involving the molecules of life – DNA. DNA consists of two strands intertwined heavily- and therefore any empirical understanding how DNA makes copies of itself will make use of Knot theory – a mathematical curiosity from the 1700s. Furthermore, in the 1960s Knot theory found applications in string theory – a very mathematical and abstract branch of physics which postulates tiny strings to be at the center of all matter.A Table of defined Knots, starting with the most basic – a circleA knotted strand of DNAThere are countless examples of old mathematical models being put to use to understand the physical world – most of them in quantum mechanics. This relationship between the physical world and mathematics is described by Physicist Mario Livio as the “passive role of mathematics” at describing nature. He distinguishes this from the “active” role which mathematics also plays; which is the fact that physicists can develop mathematical models of a system in order to understand it. These models, in turn, are so successful that it makes us question whether we are inventing, or discovering them. It is the reason Physicist Eugene Wigner published a paper entitled “the unreasonable effectiveness of mathematics in the natural sciences” in which he argues that mathematics is inherent to nature. Other modern physicists such as Max Tegmark go even further and claim that the universe itself is maths, and lays out a good argument for it (which I do not want to discuss here because I think this article is long enough as it is). An interesting point of view is voiced by mathematician Sir Michael Atiyah, in which he takes into account our brains evolutionary predisposition to make sense of the world around it:However Atiyah does recognize that this explanation does not address the “passive” effectiveness of mathematics discussed above. Even though natural selection could explain why we cope with physical phenomena on the human scale, it could not explain by mathematics successfully deals with all scales – from atoms to galaxies. Richard Hamming (1915 – 1998) believed that this could be explained by the fact that humans select and continuously improve mathematics, to fit a given situation. What some might call an evolution and human selection of mathematical ideas – a large number of ideas are spawned, but only the ones fit for describing the world survive. Even though I do believe there is truth to this – I do not see it sufficient to explain the “unreasonable effectiveness of mathematics in the natural sciences”.Is mathematics invented, or discovered?The truth is that there is no convincing answer, which makes it even more tantalizing to think about. In so many years of mathematical history the general consensus has kept changing- for all the right reasons. If we are inventing mathematics, then it is astonishing that we’ve created a seemingly ultimate tool for describing nature, and we will have to ask ourselves where for how long mathematics will remain a valid description nature – where are its boundaries? However, if we are discovering something deeper, more profound about the universe itself it would imply that mathematics is unbound, and has no end to revelations it can provide. In this case performing mathematics is equivalent to reading God’s mind.There is a middle ground, which might seem like a compromise but I believe provides valuable insights into this topic: mathematics is an intricate combination of inventions and discoveries. First, humans have to invent a concept, and declare it as such – for example prime numbers. Once this concept is declared, we can make all sorts of discoveries with it, as Euclid did when he proved that there is an infinite number of prime numbers. If one is to look at ancient Indian mathematics one will find that the concept of prime numbers had never been invented. This didn’t mean that their existence wasn’t known, it was just that the concept hadn’t been defined, making further ‘discoveries’ on that topic impossible. Another example of this is imaginary numbers; the square root of minus one. Even though this used to be a mathematical impossibility, once the concept of imaginary numbers had been defined all sorts of mathematics spawned from it – even their use in mechanics. This did not change anything about the nature of mechanics; we just invented a method for comprehending it. The reason we can even ask the question “is mathematics invented, or discovered?” is because of the consistency of mathematics. All branches are interlinked, and (most) paradoxes have been resolved over the years allowing us to fable at the consistency of mathematics. Its consistency is the reason we have so long wondered about its effectiveness, and I think the following quote by Dr. Ron Garrett helps us understand why mathematics is so effective at explaining reality.“The Universe is comprehensible because large parts of it are consistent. This consistency allows us to understand our experiences in terms of stories whose explanatory power endures from one moment to the next. (When these stories are told using mathematics we call them scientific theories.) Some of these stories, like the idea of a material object, are hardwired into the human brain. Other stories, like the idea of a chemical or electricity, are not innate. One of the triumphs of the human species is that we are able to communicate these stories, so that a new story once constructed can be propagated without having to be encoded into our DNA. Consistency defines reality. We distinguish between the perceptions that we have while sleeping from those we have while awake precisely because our wakeful perceptions are more amenable to consistent storytelling. We call our wakeful perceptions “reality” and our sleepful ones “dreams” for precisely this reason. It is so deeply ingrained in our psyche to believe that the universe is consistent because reality is in some sense real that the suggestion that reality is simply a mental construct that our brains concoct to explain consistency in perception sounds preposterous on its face. For one thing, our brains are real. If they weren’t, they wouldn’t be around to do any concocting. I will defer this issue for now; for the moment let us simply accept that consistency and reality are intimately connected without making any commitments to which way the causality runs. The point is that the Universe is comprehensible because it is consistent. This is important because comprehensibility cannot be described mathematically, but consistency can.”Mathematics gives us this consistent tool, which allows us to probe reality itself.Let me conclude my note with best finishing quotes -“Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” ― Bertrand Russell, A History of Western Philosophy"Imagine if we could look so closely we could see each grain, each particle. You see there are patterns in everything . In math, these patterns reveal themselves in the most incredible form."― Srinivasa Ramanujan"Philosophy is written in this grand book--I mean the universe--which stands continually open to our glaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it."-- Galileo Galilei (Il Saggiatore, 1623)“Besides language and music, it [mathematics] is one of the primary manifestations of the free creative powers of the human mind, and it is the universal organ for world-understanding through theoretical construction..” ― Hermann Weyl

DFW. This guy, not the airport: It's wild that many people in Silicon Valley don't recognize those initials. The other day I had to explain Infinite Jest to one of the smartest and most influential people I know. It was the first time he'd heard of the book. We could've used this literary titan, a self-deprecating genius who could tackle any subject, and he could've used us. David Foster Wallace (author): * Spoke our language. The Internet sounds like David Foster Wallace.[1] * Delighted in asking and answering questions. Like most of us, DFW had a day job — fiction, trying to tackle The Pale King — but was also a master essayist. Nonfiction was his plaything.[2] Quora could've been his playground. * Snarked. The publication: Gourmet, a prestigious food and wine magazine. His assignment: The annual(more)