Book 1, in its figures
Arithmoí Book 1 explains with pictures as much as with equations: a triangle turned over, a spiral that never closes, a spike that rises without end. Here are eight of its 207 figures, drawn moving. Each says in a few words what it shows, and leads to the page of the book that explains it: to read in the library, or to learn in the school.
Eight of the book's figures
Figure 1.1.3 · The D₃ group permutations
In part 1.1.16, The Dihedral Groups
A triangle with numbered corners can be turned and flipped onto itself in six ways: three rotations and three reflections.
Together they are the group D₃, the symmetries of the equilateral triangle, and the numbers show which move was made. The book meets it in its very first section.
Part 1.2.2 · A point turning on the unit circle
In part 1.2.2, The Cyclic Trigonometric Functions and their Derivatives
As a point turns around the unit circle, its height at each moment is the sine.
The wave is that height drawn out as the point goes round, and it comes back to where it began after one turn. It is the picture behind the cyclic trigonometric functions.
Figure 1.2.10 · The logarithmic spiral
In part 1.2.9, Numeri Mirabiles
The curve z = exp((k+i)θτ) winds around its centre without ever reaching it, and every turn is the one before it, only larger.
Jacob Bernoulli called it the spira mirabilis, the miraculous spiral, for that self-similarity. The book finds such spirals in the paths of descent of the integers.
Figures 1.3.1 to 1.3.3 · The angles of a triangle
In part 1.3.1, Curvature and Light
On a flat surface the three angles of a triangle add up to exactly π. On a sphere they add up to more, and on a pseudosphere, to less.
Adding up the angles is the book's first test of whether a space is flat, and it opens its study of curvature and light.
Figures 1.3.4 and 1.3.5 · The unit circle and the unit hyperbola
In part 1.3.1, Curvature and Light
Both curves keep one distance fixed, ρ = 1: on the circle cos²α + sin²α = 1, on the hyperbola cosh²β − sinh²β = 1.
Move the point and the triangle changes, but not the distance. The book uses the pair to ask what an angle is in a space that is not flat.
Figure 1.6.1 · The singularity
In part 1.6.1, The Singularity
For real x, the function f(x) = 1/(x² − 1) runs off to infinity at x = −1 and at x = 1: two poles.
Take x imaginary and the very same function is one continuous curve, with no poles at all. Whether something is a singularity depends on how you look at it.
Figure 1.6.19 · The Dirac delta function
In part 1.6.7, Transformations and Representation Theory
As a grows, the wave a·sinc(af) narrows and rises. In the limit it is zero everywhere except at one point, where it rises to infinity: the Dirac delta, δ(k).
The book meets it in the Fourier transform of a cosine, which is made of two such impulses.
Figure 1.7.4 · The geometric primes
In part 1.7.1, Prime Numbers and the Geometry of the Minimal Element
The primes 2, 3, 5 and 7 as regular polygons in the complex unit circle: each has as many vertices as its value.
Every polygon passes through the minimal element at 1, and every other vertex is new, one that no smaller integer's polygon reaches. The book calls them the geometric primes.
The other 199 figures are in the book
Every one of them stands in its page of Book 1, open to read in the library; the school teaches the book part by part.