Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

Geometry is 'number in space', music is 'number in time'.

In a seven-tone scale the eighth note is the octave, twice the pitch of the first note, and so signals the movement to a new level. This may be why, in religious symbolism, the eighth step is often associated with spiritual evolution or salvation.

Islamic patterns speak of infinity and the omnipresent center.

Ludwig Schlafi (1814-1895) proved that there are six regular four-dimensional polytopes (generalisations of polyhedra): the 5-cell made of tetrahedra, the 8-cell or tesseract made of cubes, the 16-cell made of tetrahedra, the 24-cell made of octahedra, the 120-cell made of dodecahedra, and the 600-cell made of tetrahedra.

Many familiar objects from cassettes to credit cards and Georgian front doors are Phi (1.618...) rectangles.

When one circle is drawn over another like this so that they pass through each others' centers, then an important almond shape, the vesica piscis, literally 'fish's bladder' is formed. It is one of the first things that circles can do. Christ is often depicted inside a vesica.

The dodecagon is also made from six squares and six equilateral triangles fitted around a hexagon

Twelve is the number which fits around one in three dimensions in the same way that six fits around one in two dimensions. The New Testament is a story of a teacher surrounded by twelve disciples.

Ten is formed from two pentagons and ten life-invoking pentagons sit perfectly arpund a decagon, and DNA, appropriately as the key to the reproduction of life, has ten steps for each turn of its double helix, so appears in cross-section as a tenfold rosette.

Eleven is important as the first number that allows us to begin to comprehend the measure of a circle. This is because, for practical purposes, a circle measuring seven across will measure eleven halfway around.

Venus draws a fivefold pattern around Earth every eight years allowing us to draw an amazing diagram. In those eight years there are almost exactly 99 full moons, nine elevens, the number of names or reflections of Allah in Islam. Jupiter draws a beautiful elevenfold pattern around Earth.

The ancient Maya were superb stargazers. Their calendar synchronized not just the Sun and Moon, byt also Venus and Mars. They worked out that 81 (or 3X3X3X3) full moons occur exactly every 2,392 (or 8X13X23) days, an astonishingly accurate gearing.

The ancients were obsessed with measures, and the number eleven is central in their metrological scheme. Shown opposite is the extraordinary fact that the size of the Moon relates to the size of the Earth as does three to eleven. What this means is that if we draw down the Moon to the Earth, as shown, then a heavenly circle through the moon will have a circumference equal to the perimeter of a square around the Earth. This is called 'squaring the circle'. Quite how the old druids worked this out we may never know, but they clearly did, for the Moon and the Earth are best measured in miles, as shown. A double rainbow also magically squares the circle.