Prime-well: strings globe

ζ built from the primes, one string at a time. Each string runs once around the globe and shows the product of the first few primes across a stretch of heights; it bulges outward wherever that product comes close to zero. The bottom string uses one prime, and each string above it uses more.

What to draw

Heights t from

Length around the globe

Primes, up to

Strings shown, at most

Lump height

Twist (turns, for looks only)

Colour the strings by

subtract the ripple (smooth allowance for the primes not yet used) mark ζ's zeros on the bright string show only the bright string and a faint trail

The globe — drag to turn · scroll to zoom · shift-drag or right-drag to move · double-click to reset

Primes used:

How to read it. Going around the globe is the height t. Going from the bottom to the top is the number of primes used. A string bulges outward where the product of its primes is close to zero, so each of ζ's zeros shows as a ridge running up the globe. The bright string is the one for the number of primes on the slider. "Play primes" lays the strings down one after another. "Play heights" keeps the primes fixed and slides the stretch of heights along, so the ridges travel around the globe. "Play length" keeps the start fixed and widens the stretch, so more and more ridges crowd in. The three can run together.

What it does and doesn't show. This is the ½ shell only. It shows the primes placing ridges where ζ's zeros are; it does not show that no other shell has a zero. A product of finitely many primes never reaches zero, so the ridges have a finite height however many strings are drawn. The twist setting is decoration: in the data each ridge stands at one fixed height.

Make a video — set what changes from the first frame to the last

Give each setting a starting and a finishing value; settings with the same value at both ends stay still. The lab draws and encodes the frames one at a time, each stamped with its own moment, so the video comes out smooth however long a frame takes to work out. The display options above (lump height, twist, colours, ripple, zeros) are used as they are.

Primes from

Primes to

Start height from

Start height to

Length from

Length to

Turn the globe (turns)

Tilt from (degrees)

Tilt to (degrees)

Zoom from

Zoom to

Length of video (seconds)

Hold last frame (seconds)

Frames a second

Picture size

Primes and length grow by equal ratios, so the early part, where most of the change happens, gets its share of the time. Heights, tilt and turning ease in and out. In browsers without a frame encoder the lab falls back to recording while the frames play, and for that the tab has to stay in front.