Philosophy [nature] is written in that great book which ever is before our eyes -- I mean the universe -- but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.

Our civil rights have no dependence on our religious opinions any more than our opinions in physics or geometry...

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.

sacred knowledge of the cosmos seems to be hidden within our souls and is shown within our artwork and creative expressions.

His way had therefore come full circle, or rather had taken the form of an ellipse or a spiral, following as ever no straight unbroken line, for the rectilinear belongs only to Geometry and not to Nature and Life.

They say the silence is the language of God, but so is music. This is why we dance, we become loud in our silence.

The Greeks made Space the subject-matter of a science of supreme simplicity and certainty. Out of it grew, in the mind of classical antiquity, the idea of pure science. Geometry became one of the most powerful expressions of that sovereignty of the intellect that inspired the thought of those times. At a later epoch, when the intellectual despotism of the Church, which had been maintained through the Middle Ages, had crumbled, and a wave of scepticism threatened to sweep away all that had seemed most fixed, those who believed in Truth clung to Geometry as to a rock, and it was the highest ideal of every scientist to carry on his science 'more geometrico.

In reality the universe has no geometry.

Maths is at only one remove from magic.

In geometry, whenever we had to find the area of a circle, pi * radius squared, I would get really hungry for pie. Square pie.

The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method—more daring than anything that the history of philosophy records—of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason.

As for methods I have sought to give them all the rigour that one requires in geometry, so as never to have recourse to the reasons drawn from the generality of algebra.

The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the Copernicus of Geometry [as did Clifford], for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.

Speed is simply the rite that initiates us intoemptiness: a nostalgic desire for forms to revert to immobility, concealed beneath the very intensification of their mobility. Akin to the nostalgia for living forms that haunts geometry.

Be honest: did you actually read [the above geometric proof]? Of course not. Who would want to? The effect of such a production being made over something so simple is to make people doubt their own intuition. Calling into question the obvious by insisting that it be 'rigorously proved' ... is to say to a student 'Your feelings and ideas are suspect. You need to think and speak our way.

I would say, if you like, that the party is like an out-moded mathematics...that is to say, the mathematics of Euclid. We need to invent a non-Euclidian mathematics with respect to political discipline.

You don’t see something until you have the right metaphor to let you perceive it

Of all the possible pathways of disorder, nature favors just a few.

Billions of years ago there were just blobs of protoplasm; now billions of years later here we are. So information has been created and stored in our structure. In the development of one person’s mind from childhood, information is clearly not just accumulated but also generated—created from connections that were not there before

the brain does not own any direct copies of stuff in the world. There is no library of forms and ideas against which to compare the images of perception. Information is stored in a plastic way, allowing fantastic juxtapositions and leaps of imagination. Some chaos exists out there, and the brain seems to have more flexibility than classical physics in finding the order in it.

Simple shapes are inhuman. They fail to resonate with the way nature organizes itself or with the way human perception sees the world.

it struck me as an operational way to define free will, in a way that allowed you to reconcile free will with determinism. The system is deterministic, but you can’t say what it’s going to do next.

In a way, art is a theory about the way the world looks to human beings. It’s abundantly obvious that one doesn’t know the world around us in detail

The only things that can ever be universal, in a sense, are scaling things.

the pattern appears so ethereally, that it is hard to remember that the shape is an attractor. It is not just any trajectory of a dynamical system. It is the trajectory toward which all other trajectories converge.

The early sense of self-similarity as an organizing principle came from the limitations on the human experience of scale.

The boundary is where points are slowest to escape the pull of the set. It is as if they are balanced between competing attractors, one at zero and the other, in effect, ringing the set at a distance of infinity.

IN THE MIND’S EYE, a fractal is a way of seeing infinity.

One simple but powerful consequence of the fractal geometry of surfaces is that surfaces in contact do not touch everywhere. The bumpiness at all scales prevents that. Even in rock under enormous pressure, at some sufficiently small scale it becomes clear that gaps remain, allowing fluid to flow.

The fractal structure nature has devised works so efficiently that, in most tissue, no cell is ever more than three or four cells away from a blood vessel. Yet the vessels and blood take up little space, no more than about five percent of the body.

The pits and tangles are more than blemishes distorting the classic shapes of Euclidian geometry. They are often the keys to the essence of a thing

Self-similarity is symmetry across scale. It implies recursion, pattern inside of pattern.

Is it possible that mathematical pathology, i.e. chaos, is health? And that mathematical health, which is the predictability and differentiability of this kind of a structure, is disease?

In the pentagram, the Pythagoreans found all proportions well-known in antiquity: arithmetic, geometric, harmonic, and also the well-known golden proportion, or the golden ratio. ... Probably owing to the perfect form and the wealth of mathematical forms, the pentagram was chosen by the Pythagoreans as their secret symbol and a symbol of health. - Alexander Voloshinov [As quoted in Stakhov]

In Euclid's Elements we meet the concept which later plays a significant role in the development of science. The concept is called the "division of a line in extreme and mean ratio" (DEMR). ...the concept occurs in two forms. The first is formulated in Proposition 11 of Book II. ...why did Euclid introduce different forms... which we can find in Books II, VI and XIII? ...Only three types of regular polygons can be faces of the Platonic solids: the equilateral triangle... the square... and the regular pentagon. In order to construct the Platonic solids... we must build the two-dimensional faces... It is for this purpose that Euclid introduced the golden ratio... (Proposition II.11)... By using the "golden" isosceles triangle...we can construct the regular pentagon... Then only one step remains to construct the dodecahedron... which for Plato is one of the most important regular polyhedra symbolizing the universal harmony in his cosmology.

Give me a place to stand, a lever long enough and a fulcrum. and I can move the Earth

I forget if it was the Mathematician of Alexandria who said that geometry is beauty laid bare or the Father of Relativity who made the claim for physics,” Darger said. “She is, in either case, ravishing.

[The golden proportion] is a scale of proportions which makes the bad difficult [to produce] and the good easy.

The Golden Ratio defines the squaring of a circle. Stated in mathematical terms, this says: Given a square of known perimeter, create a circle of equal circumference. According to some, in ancient Egypt, this mathematical mystery was encoded in the measurements of the Great Pyramid of Giza.

The geometer offers to the physicist a whole set of maps from which to choose. One map, perhaps, will fit the facts better than others, and then the geometry which provides that particular map will be the geometry most important for applied mathematics.

The play is independent of the pages on which it is printed, and ‘pure geometries’ are independent of lecture rooms, or of any other detail of the physical world.

I guess a sock is also a geometric shape—technically—but I don't know what you'd call it. A socktagon?

Is everyone with one face called a Milo?""Oh no," Milo replied; "some are called Henry or George or Robert or John or lots of other things.""How terribly confusing," he cried. "Everything here is called exactly what it is. The triangles are called triangles, the circles are called circles, and even the same numbers have the same name. Why, can you imagine what would happen if we named all the twos Henry or George or Robert or John or lots of other things? You'd have to say Robert plus John equals four, and if the four's name were Albert, things would be hopeless.""I never thought of it that way," Milo admitted."Then I suggest you begin at once," admonished the Dodecahedron from his admonishing face, "for here in Digitopolis everything is quite precise.

Everybody at the party is a many sided polygon....Nonagon!

By degrees, the bitterness at my heart diffused itself to the circumference of the circle in which my life went its cheerless mechanical round.