Projections

Stills from my interactive visualizers.

Modular arithmetic

$multiplication by 71 modulo 703$ full size · visualizer

$multiplication by 2 modulo 1000$ full size · visualizer

$multiplication by 3 modulo 1000$ full size · visualizer

$multiplication by 5 modulo 2000$ full size · visualizer

$multiplication by 51 modulo 800$ full size · visualizer

$squaring modulo 1009$ full size · visualizer

$addition of 7 modulo 30$ full size · visualizer

$multiplication by 7 modulo 200$ full size · visualizer

$multiplication by 7 modulo 199$ full size · visualizer

$multiplication by 2 modulo 200$ full size · visualizer

$multiplication by 3 modulo 200$ full size · visualizer

$multiplication by 5 modulo 200$ full size · visualizer

$multiplication by 9 modulo 200$ full size · visualizer

$multiplication by 11 modulo 200$ full size · visualizer

$multiplication by 13 modulo 200$ full size · visualizer

$squaring modulo 200$ full size · visualizer

$squaring modulo 199$ full size · visualizer

multiplication by 71 modulo 703 (dynamics) — $multiplication by 71 modulo 703$ full size · visualizer

squaring modulo 1009 (dynamics) — $squaring modulo 1009$ full size · visualizer

multiplication by 2 modulo 1001 (dynamics) — $multiplication by 2 modulo 1001$ full size · visualizer

Elliptic curves

doubling on y² = x³ + x + 1 modulo 1063 — $doubling on the elliptic curve y^{2} = x^{3} + x + 1 modulo 1063$ full size · visualizer

tripling on y² = x³ + x + 1 modulo 1009 (curve view) — $tripling on the elliptic curve y^{2} = x^{3} + x + 1 modulo 1009$ full size · visualizer

Schmidt arrangements

the Schmidt arrangement of Q(i) — $the Schmidt arrangement of Q (i)$ full size · visualizer

the Schmidt arrangement of Q(√−3) — $the Schmidt arrangement of Q (- 3)$ full size · visualizer

the Schmidt arrangement of Q(√−2) — $the Schmidt arrangement of Q (- 2)$ full size · visualizer

the Schmidt arrangement of Q(i), bold — $the Schmidt arrangement of Q (i)$ full size · visualizer

the Schmidt arrangement of Q(√−2), bold — $the Schmidt arrangement of Q (- 2)$ full size · visualizer

Divisor waves

$a wave crossing the axis at multiples of n, for each n$ full size · visualizer

$the divisors of 60$ full size · visualizer

$the divisors of 24, 36 and 60$ full size · visualizer

a wave crossing the axis at multiples of n, for each n (bold) — $a wave crossing the axis at multiples of n, for each n$ full size · visualizer

the divisors of 60 (bold) — $the divisors of 60$ full size · visualizer

the divisors of 24, 36 and 60 (bold) — $the divisors of 24, 36 and 60$ full size · visualizer

Algebraic starscapes

roots of the integer cubics ax³+bx²+cx+d — $roots of the integer cubics a x^{3} + b x^{2} + c x + d$ full size · source

roots of the integer cubics, near a root of x³+x+1 — $roots of the integer cubics, near a root of x^{3} + x + 1$ full size · source

roots of the integer quadratics ax²+bx+c — $roots of the integer quadratics a x^{2} + b x + c$ full size · source

cubic algebraic integers: roots of x³+ax²+bx+c — $cubic algebraic integers: roots of x^{3} + a x^{2} + b x + c$ full size · source

quartic units: roots of x⁴+ax³+bx²+cx±1 — $quartic units: roots of x^{4} + a x^{3} + b x^{2} + c x \pm 1$ full size · source

roots of ax³+bx+c — $roots of a x^{3} + b x + c, a, b, c \in Z$ full size · source

roots of ax³+(c+1)x²+bx+c — $roots of a x^{3} + (c + 1) x^{2} + b x + c, a, b, c \in Z$ full size · source

roots of ax³+bx²+ax+c — $roots of a x^{3} + b x^{2} + a x + c, a, b, c \in Z$ full size · source

roots of ax³+cx²+bx+c — $roots of a x^{3} + c x^{2} + b x + c, a, b, c \in Z$ full size · source

roots of ax³+bx²+cx+a — $roots of a x^{3} + b x^{2} + c x + a, a, b, c \in Z$ full size · source

roots of ax³+x²+bx+c — $roots of a x^{3} + x^{2} + b x + c, a, b, c \in Z$ full size · source

roots of ax⁴+bx²+bx+c — $roots of a x^{4} + b x^{2} + b x + c, a, b, c \in Z$ full size · source

roots of ax⁵+bx³+cx²+bx+a — $roots of a x^{5} + b x^{3} + c x^{2} + b x + a, a, b, c \in Z$ full size · source

roots of the reciprocal quartics ax⁴+bx³+cx²+bx+a — $roots of the reciprocal quartics a x^{4} + b x^{3} + c x^{2} + b x + a$ full size · source

$roots of the reciprocal sextics: p (x) = x^{6} p (1/ x)$ full size · source

$roots of the reciprocal octics: p (x) = x^{8} p (1/ x)$ full size · source

$roots of the reciprocal decics: p (x) = x^{10} p (1/ x)$ full size · source

Minkowski lattice

the circles of the lattice Z^4 in Minkowski space — $the circles of the lattice Z^{4} in Minkowski space$ full size · visualizer

Moving pictures

the circles of a lattice in Minkowski space, under a time shear

30 s loop, 1920 × 1080, silent. The lattice ℤ⁴ deformed by the time shear τ ↦ τ + s·x with s = 1.5 sin t, seen on the sphere; colour + shell at α = 2.03, σ = 1.47. · download (compressed) · visualizer

the circles of a lattice in Minkowski space, under a time shear

60 s loop, 1920 × 1080, silent. The same deformation at half speed with lines twice as thick and slightly more opaque, made for projection on a wall. · download (compressed) · visualizer

x \mapsto x^{2} modulo n, n = 125, \dots, 137

58 s loop, 1920 × 1080, silent. The dynamics of squaring on the integers mod n, morphing from one modulus to the next, 125 to 137 and back to 125; the caption follows. · download · visualizer

x \mapsto 2^{x} modulo n, n = 119, \dots, 134

72 s loop, 1920 × 1080, silent. The dynamics of x ↦ 2^x on the integers 0 … n−1, morphing from one modulus to the next, 119 to 134 and back to 119; the caption follows. Fewer points, so the segments read at a distance. · download · visualizer

x \mapsto 2^{x} modulo n, n = 1010, \dots, 1030

94 s loop, 1920 × 1080, silent. The dynamics of x ↦ 2^x on the integers 0 … n−1 (trees feeding cycles), morphing from one modulus to the next, 1010 to 1030 and back to 1010; the caption follows. · download (compressed) · visualizer