Section

Geometry

Simplexes, star polygons and the Platonic solids: the forms that unfold from the triangle of points.

1. 01Ornaments in four-four timeFill the simplexes with colour in alternation and ornaments appear whose centre is filled for four numbers in a row, then empty for four.
2. 02The simplex – the solid form of numberEvery simplex shows the complete divisibility of its number. That makes geometry a second state of matter for numbers.
3. 03Angle sums and full circlesThe square is the only polygon whose angles add up to exactly one full circle. The decagon with all its stars has exactly ten.
4. 04The simplex – multidimensional tetrahedraEach new vertex lifts the tetrahedron into a further dimension. Pascal's triangle is the complete table of these solids, and the 5-cell carries a pentagram within it.
5. 05Circle, triangle and squareHumanity's three oldest symbols are also the basis of every gap-free tiling, and the divisors of 24 decide which polygons can occur in it.
6. 06Gap-free filling of spaceTetrahedra and octahedra fill space together. Their smallest common fractal has 10 vertices and 24 edges: 1 + 2 + 3 + 4 and 1 × 2 × 3 × 4.
7. 07Kissing numbers, sphere packings and the number 242 segments around one, 6 circles around one, 12 spheres around one, 24 hyperspheres around one: the kissing numbers of the first four dimensions.
8. 08The Platonic solids and the tetractysFive perfect solids, two dual pairs and one solid that is dual to itself, assigned to the start figures of the tetractys.
9. 09The tetractys in the fractal polygonsThe Sierpiński procedure applied to n-gons. From the pentagon on the parts overlap, and in the decagon the overlaps arrange themselves as 1 + 2 + 3 + 4 = 10.
10. 10Sacred geometry – scepticism and open questionsFlower of life, merkaba and Platonic solids. What sacred geometry shares with the tetractys, and where scepticism is in order.