maths

Pictures from an exploration of the Riemann zeta function on a ball, and where it led: spirals, horizons, light cones and colour.

Plates

18 pictures, in the order they were drawn. Click one for the full-size image.

1Quarter turns
2The plane phase
3Thermographs as resting balls
4Dividing by zero is a turn
5The ball as a box kite
6The centre becomes the bottom
7Light stays light
8A pair from one point of light
9Straight in the chart, curved on the ball
10One ball, three moves, one rule
11The golden boost
12Three charts, one circle
13Touching at 0
14Opposite loxodromes
15Spirals that don't overlap
16The equator as a horizon
17The horizon collapsed to a point
18RGB, CMY and the ball

12 October 2026

ζ on a ball: turns, crushes and mirrors

The zeta ball page: the ball on the left, its shadow on the s-plane on the right

The zeta ball, to turn yourself

Drag the ball, or turn it a quarter at a time about its three axes, and watch its shadow on the s-plane. Paint it with ζ or ξ, and slide from the plane phase into ζ.

Open the zeta ball

Quarter turns: ζ painted on a ball, colour showing its phase, and seen from the floor after quarter turns about three axes: through ½ and ∞, through 0 and 1, and through ½ ± i/2. — 1Quarter turns
ζ painted on a ball, colour showing its phase, and seen from the floor after quarter turns about three axes: through ½ and ∞, through 0 and 1, and through ½ ± i/2.

The plane phase: 3Blue1Brown's continued grid under ζ, then under 1/(1 − 2⁻ˢ), whose grid is the zeta ball's own: foci 0 and 1, centre ½. Moving the foci to ±1 gives the same sphere in its plane phase. — 2The plane phase
3Blue1Brown's continued grid under ζ, then under 1/(1 − 2⁻ˢ), whose grid is the zeta ball's own: foci 0 and 1, centre ½. Moving the foci to ±1 gives the same sphere in its plane phase.

Thermographs as resting balls: ±1 and ½ ± ½ drawn as cross-sections of balls resting on a line: the lamp's rays through the poles make the walls, and the upright axis makes the mast. — 3Thermographs as resting balls
±1 and ½ ± ½ drawn as cross-sections of balls resting on a line: the lamp's rays through the poles make the walls, and the upright axis makes the mast.

Dividing by zero is a turn: On the ball, w ↦ 1/w is a half-turn, while multiplying by 0 or ∞ crushes every point to one: a zero divisor. The squish (1 + X)/2 then takes square roots of 1 to idempotents. — 4Dividing by zero is a turn
On the ball, w ↦ 1/w is a half-turn, while multiplying by 0 or ∞ crushes every point to one: a zero divisor. The squish (1 + X)/2 then takes square roots of 1 to idempotents.

The ball as a box kite: The ball's six frame points as a box kite: 0 and ∞ are the ends of a strut, and the mirror swaps sails and vents while fixing the critical circle. — 5The ball as a box kite
The ball's six frame points as a box kite: 0 and ∞ are the ends of a strut, and the mirror swaps sails and vents while fixing the critical circle.

The centre becomes the bottom: With 2×2 Hermitian matrices as Minkowski space, the squish sends the old centre to the new apex, so each sphere sits on a light cone. — 6The centre becomes the bottom
With 2×2 Hermitian matrices as Minkowski space, the squish sends the old centre to the new apex, so each sphere sits on a light cone.

Light stays light: In the chart z = 1 − 1/s the critical line is the unit circle, the Julia set of z ↦ zⁿ: orbits inside fall to 0, orbits outside run off to ∞, and orbits on the circle stay on it. — 7Light stays light
In the chart z = 1 − 1/s the critical line is the unit circle, the Julia set of z ↦ zⁿ: orbits inside fall to 0, orbits outside run off to ∞, and orbits on the circle stay on it.

A pair from one point of light: Two mirror-image log spirals leave a point of the circle, one winding into 0 and one out to ∞. In log coordinates they are two straight lines leaving the light symmetrically. — 8A pair from one point of light
Two mirror-image log spirals leave a point of the circle, one winding into 0 and one out to ∞. In log coordinates they are two straight lines leaving the light symmetrically.

Straight in the chart, curved on the ball: The log chart is Mercator's projection of the ball, so the pair are rhumb lines: straight on the map, spiralling into the two poles on the ball. — 9Straight in the chart, curved on the ball
The log chart is Mercator's projection of the ball, so the pair are rhumb lines: straight on the map, spiralling into the two poles on the ball.

13 October 2026

Spirals, horizons, light cones and colour

One ball, three moves, one rule: The turn, the crush and the mirror on one picture, tried on Weil's curve y² = x³ + x + 1 over F₅: its two zeros are the ends of one strut, and Hasse's bound is a spin rather than a boost. — 10One ball, three moves, one rule
The turn, the crush and the mirror on one picture, tried on Weil's curve y² = x³ + x + 1 over F₅: its two zeros are the ends of one strut, and Hasse's bound is a spin rather than a boost.

The golden boost: All twenty curves y² = x³ + Ax + B over F₅ have traces between −4 and 4. The traces ±5, which would break the Riemann hypothesis, would be golden boosts, with cosh in place of cos. — 11The golden boost
All twenty curves y² = x³ + Ax + B over F₅ have traces between −4 and 4. The traces ±5, which would break the Riemann hypothesis, would be golden boosts, with cosh in place of cos.

Three charts, one circle: Stereographic projection joins the flat plane, the ball and Mercator's cylinder. Their scale factors, 2/(1 + r²) and sech(log r), both equal 1 only on the light. — 12Three charts, one circle
Stereographic projection joins the flat plane, the ball and Mercator's cylinder. Their scale factors, 2/(1 + r²) and sech(log r), both equal 1 only on the light.

Touching at 0: A ball of diameter 1 resting on the plane at 0: the same rays from ∞, the light circle at height ½, and the cone of rays through it, a light cone from ∞. — 13Touching at 0
A ball of diameter 1 resting on the plane at 0: the same rays from ∞, the light circle at height ½, and the cone of rays through it, a light cone from ∞.

Opposite loxodromes: Loxodromes from both poles meeting the light at −c: the pair's mirror image, a V at c and a Λ at −c in the log chart. — 14Opposite loxodromes
Loxodromes from both poles meeting the light at −c: the pair's mirror image, a V at c and a Λ at −c in the log chart.

Spirals that don't overlap: The pair and its half-turn: two spirals with the same hand that interleave without meeting. The next version up has four, ending at ±1 and ±i. — 15Spirals that don't overlap
The pair and its half-turn: two spirals with the same hand that interleave without meeting. The next version up has four, ending at ±1 and ±i.

The equator as a horizon: The plane as the slice of an eternal black hole: the light circle is the throat, 0 and ∞ are the two far-away ends, and the mirror swaps the sides while the clock runs backwards on the other side. — 16The equator as a horizon
The plane as the slice of an eternal black hole: the light circle is the throat, 0 and ∞ are the two far-away ends, and the mirror swaps the sides while the clock runs backwards on the other side.

The horizon collapsed to a point: Pinch the hourglass |w|² − t² = a² and it becomes the light cone, with the throat as the apex. A massive path bounces and keeps its side; light goes straight through to the opposite side. — 17The horizon collapsed to a point
Pinch the hourglass |w|² − t² = a² and it becomes the light cone, with the throat as the apex. A massive path bounces and keeps its side; light goes straight through to the opposite side.

RGB, CMY and the ball: Each half-turn partner is the sum of the other two (−1 = ω + ω²), so the cube roots and their half-turns are the RGB cube seen down its diagonal. On the colour ball, the antipode is the photographic negative. — 18RGB, CMY and the ball
Each half-turn partner is the sum of the other two (−1 = ω + ω²), so the cube roots and their half-turns are the RGB cube seen down its diagonal. On the colour ball, the antipode is the photographic negative.