Cosmopoeia
Sections

Section

Cosmos

What modern science finds about number and form: resonances and Trojans in the sky, quasicrystals, atomic shells and the light of hydrogen, plants, viruses and honeycombs, and sounds that draw figures.

Sky

  1. 01Resonances – the numerical ratios of the heavenly bodiesIo, Europa and Ganymede orbit Jupiter in the ratio 1 : 2 : 4, the planets of TRAPPIST-1 form a chain of fifths and fourths. The harmony of the spheres in today's astronomy.
  2. 02Is the universe a dodecahedron?Plato assigned the dodecahedron to the whole. In 2003 cosmologists proposed that space itself might have the shape of a dodecahedron, based on the cosmic background radiation.
  3. 03The Trojans – the equilateral triangle in the skyThe Sun, Jupiter and thousands of asteroids form equilateral triangles. Lagrange found this solution in 1772, the first Trojan was discovered in 1906.

Matter

  1. 04Quasicrystals – the forbidden fiveCrystals cannot have fivefold symmetry, crystallography taught. In 1982 Dan Shechtman found it nonetheless. The order behind it is that of the Penrose tiling and the golden ratio.
  2. 05Sphere packings in 8 and 24 dimensionsHow densely can spheres be packed? In three dimensions Hales proved it in 1998, in eight and 24 dimensions Maryna Viazovska in 2016. There lie the E8 lattice and the Leech lattice, and the number 24.
  3. 06Atomic shells and magic numbersElectron shells hold 2, 8, 18, 32 electrons, twice the square numbers. The first magic numbers of atomic nuclei, 2, 8 and 20, are twice the tetrahedral numbers 1, 4 and 10.
  4. 07The Balmer lines – ratios in lightThe light of hydrogen has four visible colours. In 1885 Johann Jakob Balmer found their wavelengths as simple fractions of square numbers, among them 4 : 3 and 9 : 8, the fourth and the whole tone.

Life

  1. 08The golden angle – how plants countSunflower seeds, pine cones and leaf arrangements grow 137.5° apart, the golden angle. Spirals appear in Fibonacci numbers, 21, 34, 55.
  2. 09Viruses and fullerenes – Platonic solids in natureMany viruses build their shell as an icosahedron, and the carbon molecule C60 is a football of 12 pentagons and 20 hexagons. It is named after Buckminster Fuller.
  3. 10The honeycomb – why hexagonsOnly triangles, squares and hexagons fill the plane without gaps, and the hexagon needs the least wall. Suspected since antiquity, proven in 1999.

Sound and form

  1. 11Chladni figures – sound that draws formsScatter sand on a vibrating plate and it gathers into symmetrical patterns. In 1787 Ernst Chladni made sound visible, and the patterns follow whole numbers.