P-adic wheels

In the usual (real) number line, two numbers are “close” if their leading digits agree: ${3.14159}$ is closer to ${3.14160}$ than to ${3.2}$ . The p-adic world flips that intuition: closeness is governed by agreement of the low-order digits (divisibility by powers of a prime).

This post introduces a small Pygame visualization (shown as circles inside circles inside circles) for the 3-adic integers ${\mathbb{Z}_3}$ . The picture is inspired by the beautiful illustrations of Heiko Knospe, and the app is meant to make two ideas feel tangible: (1) what it means for numbers to be p-adically close, and (2) how addition behaves in this nested, ultrametric geometry.

Hensel’s “invention” of the p-adic numbers

The p-adic numbers were first systematically introduced by Kurt Hensel (1861–1941), a German mathematician who studied in Berlin and Bonn and was strongly influenced by Leopold Kronecker; from 1901 onward he was a professor in Marburg. Around 1897, Hensel realized that one can build a number system by completing the rationals not with the usual absolute value, but with a new absolute value ${|\cdot|_p}$ that measures divisibility by a fixed prime ${p}$ .

A compact way to say “p-adic closeness” is: two integers ${x}$ and ${y}$ are close in the p-adic sense if ${x-y}$ is divisible by a large power of ${p}$ . Write ${v_p(x-y)}$ for the exponent of ${p}$ in ${x-y}$ . Then ${|x-y|_p = p^{-v_p(x-y)}}$ is small exactly when ${v_p(x-y)}$ is large.

If you’d like a classic reference from Hensel himself, see his book Zahlentheorie (1913), where p-adic methods appear in a mature, systematic form.

From congruences to nested circles

The 3-adic integers ${\mathbb{Z}_3}$ can be viewed as an infinite limit of congruence systems, and that viewpoint is perfect for visualization. A basic “clopen” (closed-and-open) 3-adic ball has the form ${a + 3^n\mathbb{Z}_3}$ , which you can read as “all 3-adic integers that are congruent to ${a}$ modulo ${3^n}$ .”

In the app, those balls become nested circles:

The big circle is the whole space (schematically).
Inside it are 3 circles: the residue classes mod 3 (the least significant ternary digit ${d_0\in\{0,1,2\}}$ ).
Inside each of those are 3 circles: refining to mod 9 (digit ${d_1}$ ).
Inside each of those are 3 circles: refining to mod 27 (digit ${d_2}$ ).
And one more layer gives mod 81 (digit ${d_3}$ ), for a total of ${3^4=81}$ innermost circles labeled 0–80.

Two numbers land in the same small circle precisely when they agree modulo a high power of 3. That’s the visual meaning of “3-adically close”: sharing deeper circles means sharing more low-order ternary digits, i.e., ${x\equiv y\ (\mathrm{mod}\ 3^n)}$ for larger and larger ${n}$ .

Addition as a ternary wheel

The app animates the operation ${x\mapsto x+1}$ (wrapping ${80\mapsto 0}$ ), and highlights (i) the active innermost circle and (ii) every circle that contains it. Conceptually, this is the “wheel” behavior of ternary digits: the last digit ticks 0→1→2→0, and whenever it rolls over, a carry moves to the next digit.

The nesting makes carries feel geometric. When a carry happens, the highlight doesn’t merely move to a nearby leaf; it “jumps” to a different region determined by the next digit layer. In p-adic terms, carries change higher digits, while leaving lower congruence information intact.

Here is a simple invariant you can read directly from the circles: adding 3 does not change the residue mod 3, because ${3\equiv 0\ (\mathrm{mod}\ 3)}$ . In the visualization, that means ${x}$ and ${x+3}$ stay inside the same one of the three outer circles.

Likewise, adding 9 does not change the residue mod 9, because ${9\equiv 0\ (\mathrm{mod}\ 9)}$ . So ${x}$ and ${x+9}$ keep the same position in the second layer (the mod 9 refinement), even though they may move around in deeper layers.

Why this picture is helpful (and what it is not)

This is not an isometric embedding of ${\mathbb{Z}_3}$ into the Euclidean plane. It is a schematic drawing that encodes the key ultrametric fact: balls are nested, and each ball splits cleanly into three disjoint sub-balls at the next scale.

If you want to generalize, the same idea works for any prime ${p}$ : each ball splits into ${p}$ sub-balls, and p-adic closeness is agreement modulo ${p^n}$ . The circles then become a compact way to “see” both topology (nested clopen sets) and arithmetic (addition with carries).

Credits & references

Visualization idea inspired by the illustrations of Heiko Knospe.
K. Hensel, Zahlentheorie, 1913.
P-adic number wheel animation: <a href=”https://trinket.io/library/trinkets/71ecc4abf5fa” Link to Interactive Python Code

Michael Emmerich

P-adic wheels

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P-adic wheels

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