The collection

Every essay — page 3

Page 3 of 44, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Series Concepts Search

Geometry

Shapes, and the arguments that can be made by rearranging them.

100 as three triangular numbers. 100 drawn as three triangles of dots with 36, 36 and 28 dots. There are 6 such decompositions.

Three triangular numbers, and no fewer

A triangle, its midpoints and its centroid, turned into lines. The dual arrangement of 7 points: one line per point, crossing where points were collinear. 3 crossings are of exactly two lines, the dual of the ordinary lines; the others are where three or more meet.

Three ordinary lines from a count

Böröczky's 12 points and their 6 ordinary lines. A disc standing for the projective plane: the 6 corners of a regular polygon inside, and 6 points at infinity marked in pairs on the rim. All 22 connecting lines are drawn, the 6 ordinary ones solid.

The fewest ordinary lines a polygon allows

Every chord through the point cuts into pieces whose product is 16.00. A circle with a point inside it and four chords drawn through the point, each labelled with the lengths of its two pieces, beside four rectangles whose sides are those pieces and whose areas are all equal.

One number for every chord through a point

Each side over the sine of the opposite angle is the diameter. A triangle inscribed in a circle, with the diameter from one corner drawn and joined to a second corner to make a right-angled triangle; the angle opposite the side at the far end of the diameter equals the triangle's own angle opposite that side.

Every side measured by one diameter

Conway's seven pieces, pulled apart and measured. A triangle with angles 78, 54, 48 degrees divided along its angle trisectors into an equilateral centre, three pieces touching it along its sides and three along the outer sides, spread apart, with each corner's angle labelled as a third of an outer angle plus a multiple of sixty degrees.

Seven pieces and an equilateral middle

Twenty-seven choices of trisector, and the eighteen equilateral triangles. Twenty-seven small panels, one for each choice of trisecting line at each corner of a triangle, each drawing the triangle and the triangle the chosen lines cut out; the eighteen equilateral ones are marked.

Eighteen equilateral triangles

Euclid's game on 34 and 21. A 34 by 21 rectangle tiled by the squares of Euclid's algorithm, each run of equal squares shaded by the player who faces it, with the deciding run outlined.

The player who meets the first long run

Counting the real roots of x⁵ − 5x³ + x² + 3x − 1 with Euclid's algorithm. Above, the graph of a polynomial over an interval with its real roots marked; below, a step function counting sign changes in the polynomial's Sturm chain, which drops by one at each root; beside them the chain of polynomials listed.

The remainders that count the roots

A pentagon, its pentagram, and the pentagon inside. A regular pentagon with its diagonals drawn as a pentagram, enclosing a smaller pentagon, repeated 3 times inward; every pentagon has diagonal-to-side ratio φ.

The diagonal no unit measures

A patch of Penrose's rhombs. A disc of Penrose rhombus tiling after 5 subdivisions, thick and thin rhombs shaded differently: 550 thick and 340 thin half-rhombs.

Tiles that never repeat

Three tilings by cells of one area, and the wall each needs. Panels of triangles, squares and hexagons, all with cells of the same area, labelled with the wall length each cell needs once shared walls are split between neighbours: the hexagons need the least.

The least wall for equal rooms

One symmetrisation: every chord slid to the middle. A lopsided shape with a dent beside its Steiner symmetrisation about a horizontal line, with a few vertical chords marked in both: the chords keep their lengths and are centred on the line.

Every chord slid to the middle

Removing one end of an ordinary line from a triangle, its midpoints and its centroid. Two panels. Left: 7 points with all 9 connecting lines, one ordinary line solid and one of its ends ringed. Right: the same points with that end removed, 7 connecting lines left.

At least as many lines as points

9 trees on a cubic, 10 rows of three. The curve y = 1/(1 + x²) with 8 marked points on it and a 9th at infinity. Every line through three of them is drawn: 10 in all.

Rows of three, planted on a cubic

The Voronoi diagram of points on a shape's boundary, and the axis it traces. 50 boundary samples of a shape with their Voronoi cells clipped to the shape; the cells' interior corners line up along the medial axis.

The skeleton inside a shape

The 9 mirrors of the octahedral group, cutting the sphere into 48 triangles. A sphere crossed by the 9 great circles of the octahedral group's mirror planes, dividing it into 48 triangles with one shaded.

Three mirrors make every solid

Three lines through a point, reflected in the bisectors, meet again. A triangle with a point P and its cevians, their reflections in the angle bisectors, and the point P* where the reflections meet — P's isogonal conjugate, at (373.8, 290.0).

Every point has a partner across the bisectors

Five triangles between the same two circles, every one closing. An outer circle of radius 1 and an inner of radius 0.38 at Euler's distance 0.4899; five triangles inscribed in the first and circumscribed about the second, started from different points.

A triangle that fits once fits everywhere

Bands of a sphere and of its cylinder, cut by the same planes. A unit sphere inside its cylinder with 3 horizontal bands; each sphere band has area 1.885, 1.885, 1.885, equal to the cylinder band of the same height.

A band of a sphere is a band of its cylinder

The volume of the unit ball and the area of its sphere, dimension by dimension. Unit ball volumes for dimensions 0 to 20, largest at 5 (5.2638); sphere areas largest at 7 (33.0734).

The ball that is largest in five dimensions

The fewest pentagonal numbers adding to each number up to 120. Fewest pentagonal numbers summing to 1..120; the most ever needed up to 20000 is 5, by 9, 21, 31, 43, 55, 89.

Six numbers that need five pentagons

The fifty-four sums of four weighted squares that reach every number. The 54 universal diagonal quaternary forms, as coefficient lists: 1,1,1,1; 1,1,1,2; 1,1,1,3; 1,1,1,4; 1,1,1,5; 1,1,1,6; 1,1,1,7; 1,1,2,2; 1,1,2,3; 1,1,2,4; 1,1,2,5; 1,1,2,6; 1,1,2,7; 1,1,2,8; 1,1,2,9; 1,1,2,10; 1,1,2,11; 1,1,2,12; 1,1,2,13; 1,1,2,14; 1,1,3,3; 1,1,3,4; 1,1,3,5; 1,1,3,6; 1,2,2,2; 1,2,2,3; 1,2,2,4; 1,2,2,5; 1,2,2,6; 1,2,2,7; 1,2,3,3; 1,2,3,4; 1,2,3,5; 1,2,3,6; 1,2,3,7; 1,2,3,8; 1,2,3,9; 1,2,3,10; 1,2,4,4; 1,2,4,5; 1,2,4,6; 1,2,4,7; 1,2,4,8; 1,2,4,9; 1,2,4,10; 1,2,4,11; 1,2,4,12; 1,2,4,13; 1,2,4,14; 1,2,5,6; 1,2,5,7; 1,2,5,8; 1,2,5,9; 1,2,5,10.

Fifteen numbers decide every number

The proof of Descartes' rule, one term at a time. Four stacked graphs: a four-term polynomial with three positive roots, and the three polynomials obtained by dividing out a power of x and differentiating, each with one fewer term, one fewer sign change and one fewer positive root.

What the signs allow