Quadrivium Quotes

“Geometry is 'number in space', music is 'number in time'.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“We see that music, like the world, is formed from unchanging mathematical principles deployed in time, creating complexity, variety and beauty.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“Islamic patterns speak of infinity and the omnipresent center.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“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.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“Music is carried by the vibrations of molecules of air, like waves upon an ocean. It perhaps uniquely captures and conveys the interior landscape of one human mind to another, holding our tears and sweat, pain and pleasure, packaged as paeans and preludes and etudes and nocturnes. It is the texturization of the deliquescence of time, the ebb and flow of mood and meaning. It ruminates, vacillates, contemplates, and stimulates.

In music we organize and fantasize, arranging the elements of music-melody, rhythm, and harmony-into meaningful shapes and patterns. Its rhythms move our hands, feet and bodies to the pulses of the universe. Its harmonies breathe with the exploratory intricacies and curiosities of relationship and proportion, consonance, dissonance, assonance, and resonance. Its melodies flitter into flights of fancy, weaving woe and wonder.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“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.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“A life's journey is like a musical composition, born into the world from nothing, living for a time in form and structure, dancing spontaneously on the edge of chaos and order, and then finally returning. In this respect Western music tends to be more linear, Eastern music more cyclical.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“Music shapes and shivers into endless colours, nuanced and diverse, and eternally creative. It is Spirit taking form.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“It's strange. With all this harmonious interplay of numbers you would have expected the whole system to be a precisely coherent whole. It isn't. There are echoes here from the scientific view of a world formed by broken symmetry, subject to quantum uncertainty and (so far) defying a precise comprehensive 'theory of everything'. Is this why the 'near miss' is so often more beautiful than perfection?”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“If you kept on spiraling you would eventually discover, as the Chinese did long ago, that 53 perfect fifths (or Lu) almost exactly equal 31 octaves. The first five fifths produce the pattern of the black notes on a piano, the Eastern pentatonic scale.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“Schlafi also proved that in five or more dimensions the only regular polytopes are the simplex, or generalized tetrahedron, the hypercube, or generalized cube, and the orthoplex, or generalized octahedron.”
― John Martineau, Quadrivium: Number Geometry Music Heaven

“The rhombic dodecahedron is a three-dimensional shadow of the four-dimensional tesseract analogous to the hexagon as a two-dimensional shadow of the cube.”
― John Martineau, Quadrivium: Number Geometry Music Heaven

“Many familiar objects from cassettes to credit cards and Georgian front doors are Phi (1.618...) rectangles.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“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.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“The dodecagon is also made from six squares and six equilateral triangles fitted around a hexagon”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“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.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“The sphere is a symbol of the cosmos and the totality of manifest creation. Very large and very small things in nature tend to be spherical. Einstein discovered that a point in four dimensions (i.e. you here and now) is a sphere expanding at the speed of light, and all we can see of the entire universe is inside an event-horizon sphere. The cube represents the earth.”
― John Martineau, Quadrivium: Number Geometry Music Heaven

“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.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“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.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“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.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“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.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology

“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.”
― John Martineau, Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology