Penrose figure: Where B13 Started, and the Figure the AI Couldn't Draw

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It started on paper

![Hand-drawn arrangement of ten pentagons forming a ring, central portion]
Fig. 1 — Ten pentagons laid out by hand.

B13 did not start from a theorem or a conjecture. It started from a figure: ten pentagons arranged in a ring.

Lay ten regular pentagons edge to edge in a closed loop and a hole opens in the middle. Line several of these rings up and a pentagram stands at the centre. Making that the core of a processor — that idea was there from the beginning. In February 2026 I confirmed it by hand and in CAD. The realisation that the ring closes in balanced quinary — where stepping back two becomes the addition of a negative digit — also happened on paper, not with an AI.

From there came the residue, then the landing, then the address, and eventually the B13 of today. Having gone as deep into number theory as this has, it is easy to forget: the material was ten pentagons from the start.

Note on vocabulary: residue, landing and address are terms of this framework, not the standard number-theoretic ones. They are defined in a later post in this series. Nothing here is reduced modulo anything.

The figure the AI could not draw

By hand, this structure goes forward. Probably forever.

An AI could not draw it. The generated figures broke down partway through, the connections could not be turned into numbers, and the pentagons came out either overlapping or full of gaps. Without an exact path that can be sent to a printer, nothing moves. That was the limit of what AI could do at the time, and the matter was closed.

This time I put the same question again. What follows is the record of that attempt.

Some of the failures:

AI is bad at seeing shapes

The first thing that turned up was that the AI's own definition of the ring was wrong.

It had been setting the ring radius so that the centre-to-centre distance between neighbouring pentagons equalled the edge length. But when two pentagons share an edge, the distance between their centres is not the edge length. It is twice the inradius.

Observer's note — labelled as borrowed knowledge:
For a regular pentagon with circumradius 1, the inradius is

$$\cos(\pi/5)$$

. If an edge is shared, the centre-to-centre distance is

$$2\cos(\pi/5) = \varphi$$

.
For ten of them in a ring, the ring radius is

$$\varphi / (2\sin(\pi/10)) = \varphi^2$$

.

While that error was in place, 20% of each generated pentagon's area was overlapping its neighbour. They were being laid down on top of each other, so of course the figure could not be continued.

The correct ratio was confirmed against an earlier broken figure — the one with the gaps — by measuring it in pixels. With a correction for the cell area coming out small by the width of the drawn line, the centre-to-centre distance inside the ring matched 1.618.

So the correct ring is:

• ten pentagons, neighbours sharing an edge
• ring radius
  $$R = \varphi^2$$
• the hole at the centre is a regular decagon

There are only two ways to connect

I had every arrangement of two rings enumerated, with overlap forbidden. The result was clean.

There are exactly two ways two rings can be placed without overlapping.

Pattern	Pentagons shared	Ring centre distance
Adjacent sharing	two consecutive	4.98
Skip sharing	two with one skipped between	3.078

At any other distance or angle, some pentagon always overlaps another. Arrangements touching at a single point do exist, but they open the structure up too far to be a structure.

That there are only these two was something I had already established by hand. The AI's exhaustive enumeration only confirmed it by machine. The attribution is stated here explicitly.

Do not put a ring at the centre

There was a second error on the AI's side: it had been putting a ring at the centre.

One ring in the middle, five more around it at 72° intervals. Brute-forcing every distance and every rotation gives zero valid arrangements. It is geometrically impossible.

The core is not a ring. The core is a pentagram. The moment the central ring was removed, the five rings fell into place. The original description — arrange the rings around a pentagram — had been correct as written. (Fig. 4, right.)

And a second figure appears alongside it: the tightly packed version (Fig. 4, left).

The first layer comes in two kinds

The first layer, made of five rings, can be built two ways.

Arrangement	Ring centre distance	Pentagons	Shared
A	$$\varphi^2$$	36	14
B	$$\varphi^3$$	40	10

Both have zero overlap and five-fold rotational symmetry. Both are correct.

Each additional layer needs five more rings: 5, 10, 15. The second layer's ten rings trace a pentagon whose vertices point the opposite way from the first layer's, offset by 36°. That is where both adjacent sharing and skip sharing become necessary. Once the meaning of those two patterns can be read, the second layer and everything after it should follow on its own.

Writing it with integers only

The drawing code is written in integer arithmetic alone. Floating point appears in exactly one place: handing coordinates to the SVG at the very end.

Coordinates are held as elements of Z[ζ], with ζ = e^{iπ/5} (36°), whose minimal polynomial is

$$\zeta^4 - \zeta^3 + \zeta^2 - \zeta + 1 = 0$$

An element is an integer 4-tuple:

u = [a0, a1, a2, a3]  =  a0 + a1·ζ + a2·ζ² + a3·ζ³

Multiplying by ζ, once reduced with ζ⁴ = ζ³ − ζ² + ζ − 1, is nothing but a permutation of integers with additions and subtractions:

def z_mul_zeta(u):
    a0, a1, a2, a3 = u
    return (-a3, a0 + a3, a1 - a3, a2 + a3)

φ lives inside this ring as an integer:

φ  = ζ + ζ⁻¹ = 1 + ζ² − ζ³   →  [1, 0, 1, −1]
φ² = φ + 1                    →  [2, 0, 1, −1]
φ³ = φ² + φ                   →  [3, 0, 2, −2]

Rewritten in balanced base-φ (digits −1, 0, 1), φ² is 100 and φ³ is 1000. The two kinds of first layer differ by exactly one digit position in the ring centre distance.

The generating rules fit in four lines:

ring radius            R = φ²
k-th ring of layer 1   centre = D·ζ^(2k),  rotation = 2k
its j-th pentagon      centre = D·ζ^(2k) + R·ζ^(2k+j),  orientation = 2k+j
i-th vertex            = centre + ζ^(orientation + 2i)

Shared pentagons are identified by exact equality of integer coordinates. No tolerance argument enters anywhere. Orientation is compared mod 2, since a pentagon is invariant under 72° — that is, under two address steps.

Verification: the centre-to-centre distance between neighbouring pentagons within a ring was exactly φ at all ten positions, and the maximum overlap between any two pentagons was exactly 0.

Attribution

Separating out where each claim came from.

Stated by the author

• the unit itself — ten pentagons in a ring
• that there are only two ways to connect: adjacent sharing and skip sharing
• that single-point contact exists but opens the structure too far
• that each layer adds five rings, and that the second layer is offset 36° with its vertices reversed
• the ruling that both kinds of first layer are correct

Computed by the AI

• deriving the ring radius φ² and the edge-sharing distance φ
• mechanical confirmation of "only two" by exhaustive enumeration with overlap forbidden
• confirming that zero arrangements exist with a ring at the centre
• the drawing code using the integer representation in Z[ζ]

Nothing here originated on the AI side. It brought in a wrong definition of the ring, put a ring at the centre, and offered pentagon packings that have been known since Dürer — and was sent back each time. This article is the record of a machine catching up to a structure that had already been settled by hand, and putting it into coordinates.

What comes next

A different Penrose figure. It will turn up again later.

The question I am working on is whether Penrose tiles can be made into a processor that computes by structure.