Sources
About this site
This site explores the Pythagorean tetractys in geometry, number theory, harmonics, mysticism and philosophy. Its motto is: Number theory + geometry = philosophy.
Basis
The ideas, theses and structure go back to the German website tetraktys.de by Holger Ullmann, which has existed since 2010. The texts here are newly written. The graphics were recomputed from the underlying mathematics and made interactive wherever that made sense.
How the site is meant
These pages are not meant to teach a syllabus but to encourage readers to experience symmetry for themselves. That only works hands-on: drawing simplexes, entering the figures in the division table, building models of the Platonic solids and of sphere packings and grasping them in the literal sense. In this view the primes too are pure geometry. Read little and think a lot for yourself, is the advice.
The motto means joining logical-analytical thinking (numbers as quantities) with pictorial-intuitive seeing (geometry) into a holistic understanding (philosophy), the art of networked thinking. Behind it lies the conviction that these relations are no invention of the human mind. Symmetry, balance as a law of nature, the geometry of circle and sphere, has always existed. Understanding it is one of the noblest tasks of thought.
The working rules: never lose sight of the whole, no hasty conclusions, and always be on guard against mere number games. Thinking along and researching along are expressly welcome.
How the ideas came about
Holger Ullmann recorded on his site how he arrived at his theses. In brief:
- 1970 to 1975: leaded windows with bullseye panes awaken an interest in the simplest geometric laws. Three decades follow with the feeling of an unfinished task.
- from 2005: intensive study of the crystal lattices of sphere packings and regular figures and their number-theoretical counterparts.
- 2006: filling the areas between the diagonals of a simplex alternately reveals 1 + 2 + 3 + 4 = 10. The cause remains unclear for a long time.
- January 2010: the cause lies in the angle sums of polygons and star polygons. Shortly after: star polygons made of star polygons begin in the decagon with 10 : 4, and the simplex consists of four basic types.
- March and April 2010: a rhythm of six and a rhythm of ten in the primes, and the insight that from 25 on the non-primes are sieved out by the star polygons in groups of four.
- June 2010: a mathematician confirms the geometric counterpart to the distribution of primes, and a computer program checks it up to one million.
- October 2010: discovery of a counter-running grid to the ordering grid of the primes.
- November 2010: tetraktys.de goes online.
- 2011: fanning out the simplexes into a coordinate system turns out to be directly related to the lambdoma of harmonics, and the lambdoma necessarily falls into four equal regions, in music theory as in number theory. A mathematician and physics teacher who is a friend helps settle open questions and dies in an accident shortly afterwards. A reader points out parallels to Giordano Bruno’s De monade, numero et figura.
- January 2013: discovery of Speusippus’ fragments on the Pythagorean numbers. They confirm that the Pythagoreans understood the ten in exactly this sense (see The ancient sources).
- 2013 to 2015: the Association of Masonic Arts honours tetraktys.de as one of its three websites of the year, dedicated to the geometry within the Craft. A reader recommends divisorplot.com, another notices the ten-pointed star in the cross-section of DNA. Many pages are expanded, among them Kabbalah, Platonic solids, space filling and sacred geometry. The last major revision dates from August 2015.
- since March 2021: tetraktys.de also on Instagram.
Images
All historical illustrations are in the public domain and come from Wikimedia Commons. The origin and licence of each image are listed under Image credits. All other graphics, from the triangles of points to the simplexes and the 3D models, are original constructions.
Literature
The most important books and further websites are collected under Links and media.
Technology
The site is built with Astro. The 3D models use three.js, all other graphics are SVG or canvas and are computed in the browser. The tones in the lambdoma and on the monochord are generated by the browser itself.