# An Illustrated Theory of Numbers

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The prime numbers between 1 and 1 billion are visualized at successively larger scales. The left column displays the primes from 1 to 100 as individuals. The small-scale irregularities smooth out as one considers primes between 1 and 1000, 1 and 10000, etc. The rightmost column displays the primes from 1 to 1 billion as an almost uniform shade of gray. The amount of black ink is precisely the density of primes, e.g., if 5% of numbers are prime within a range, that range is shaded by black ink at a density of 5%. Created as a vector graphic, and printed at high-resolution for crisp lines at large size. A size of 20" by 15" looks good on the wall.

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Within the Gaussian integers Z[i] and Eisenstein integers Z[w] (w a primitive cube root of unity), the Gaussian and Eisenstein prime numbers are displayed. Almost 100,000 primes are displayed in each circle, and the "prime angles" are marked with ticks around the circumference. The green pie-pieces are fundamental domains, breaking the symmetries given by the Gaussian and Eisenstein units, together with complex conjugation. A size of 24" by 15" looks good on the wall.

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The numbers 1 through 36 are arranged in 36 different diagrams, displaying the dynamics of multiplication modulo 37. Primitive roots modulo 37 appear as large circles. Each cycle is labeled by a sign (+/-) according to its sign as a permutation. This illustrates Zolotarev's Lemma, connecting the sign of the "multiplication by a mod p" permutation to the Legendre symbol. Students will find their own patterns. Who knew that a cyclic group could be so pretty? A size of 22.75" by 24" looks good on the wall, though bigger might be better if you want to see the numbers from afar.