Matpic
mathematics, in pictures
Concept

Regular polygon

A closed figure whose sides are all the same length and whose angles are all the same size.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

3 · 4 · 5 · 6 · 8 · 10 · 12 · 15 · 16 · 17 · 20 · 24 · …3φ24φ25φ46φ27φ68φ49φ610φ411φ1012φ413φ1214φ615φ816φ817φ1618φ619φ1820φ821φ1222φ1023φ2224φ825φ2026φ1227φ1828φ1229φ2830φ831φ3032φ1633φ2034φ1635φ2436φ1237φ3638φ1839φ2440φ1641φ4042φ1243φ4244φ2045φ2446φ2247φ4648φ1649φ4250φ2051φ3252φ2453φ5254φ1855φ4056φ2457φ3658φ2859φ5860φ1661φ6062φ3063φ3664φ3265φ4866φ2067φ6668φ3269φ4470φ2471φ7072φ2473φ7274φ3675φ4076φ3677φ6078φ2479φ7880φ3281φ5482φ4083φ8284φ2485φ6486φ4287φ5688φ4089φ8890φ2491φ7292φ4493φ6094φ4695φ7296φ3297φ9698φ4299φ60100φ40n = 3 to 100: 24 constructible, 74 notdecided twice — by the Fermat-prime criterion and by φ(n) being a power of two — and thetwo agreed at every one of the 98

Which polygons can be drawn

Three sides yes, seven no, seventeen yes. The list of constructible regular polygons is neither everything nor almost nothing, and the pattern in it is a fact about which numbers are one less than a power of two.

computation · constructible numbers 123412431324134214231432213421432314234124132431312431423214324134123421412341324213423143124321all 24 ways of relabelling the 4 corners, and the 8 that move no distancethe other 16 change at least one distance between corners, so no motion of the plane performs them

Eight ways to leave a square alone

A square can be picked up and put back so that nothing looks different. There are exactly eight ways to do it, and the number is not asserted here — it is what a search through all twenty-four relabellings of the corners comes back with.

algebra · symmetry groups 3 sidesarea 4,3304πA/L² = 0.6054 sidesarea 5,6254πA/L² = 0.7856 sidesarea 6,4954πA/L² = 0.90712 sidesarea 6,9984πA/L² = 0.977the circlearea 7,1624πA/L² = 1every shape here has a perimeter of 300; only the area changes4πA/L² rises from 0.605 to 0.977 and reaches 1 only at the circle

The most area a fence can hold

One length of boundary, and the question of what shape to bend it into. The answer is a circle, everybody knows it, and the argument that convinced the nineteenth century turned out to prove something slightly different.

geometry · isoperimetric

Named alongside it

The objects these essays reach for when they reach for this one.

Cyclic groupSymmetryAreaCircleClosureConstant widthConstructible numberConvexityCounting argumentDegree of an extensionDihedral groupExistence proof

All concepts