Matpic
mathematics, in pictures
Theme

Things that cannot be done — page 2

Results that close a door rather than open one — and the peculiar difficulty of drawing a picture of something that does not exist.
π = 3.141592654…coefficients from −5 to 5degreeclosest missvalue there1−x + 30.142110 tried2−2x² + 5x + 40.03121,210 tried3−x³ + 2x² + 2x + 50.016113,310 tried4−2x⁴ + 5x³ + 5x² − 4x + 30.00515146,410 tried161,040 integer polynomials of degree ≤ 4 with coefficients in [−5, 5], evaluated at π —none is zerothe same search finds x² − 2 for √2, so its silence about π is a report and not a proof Computation

The circle that will not square

The other three impossibilities are a number having the wrong degree. This one is a number having no degree at all — and that is a claim no finite search can establish, which makes it the one place in this field where the picture has to admit what it is not doing.

 +01αα+101αα+101αα+110α+1ααα+101α+1α10×01αα+101αα+1000001αα+10αα+110α+11α× mod 40000012302020321the 4 elements of GF(4), built as polynomials in α modulo α+1 + α² — every product of two non-zero elements is non-zerobeside it the integers mod 4, where 2 × 2 = 0 and 1 element has no reciprocal Computation

The field with four elements

The integers modulo four are not a field: two times two is zero and two has no reciprocal. There is nevertheless a field with four elements, and building it means giving up on counting as the way to make arithmetic finite.

 order 6, symbols 0…5012345123450234501345012450123501234transversals of the cyclic squareorder 33order 40order 515order 60order 7133the cyclic square of order 6 has no transversal at all: all 720 placements were tried and every one repeats asymbola square with no transversal cannot have an orthogonal mate, so the search for one need never begin Computation

The thirty-six officers

Six regiments send six officers each, one of every rank. Arrange all thirty-six in a square so that each row and each column holds every rank once and every regiment once. Euler could not, guessed why, and was wrong about the reason.

All themes