# ♢ ⧫ ⬠: the fourth kind of Penrose tiling?

It’s known that Penrose tilings have several implementations that are mutually locally derivable; but the sources (such as en.wikipedia) list no more than three essentially different variants. There is no difficulty in inventing yet another one by modifying tiles’ shape (and possibly rearranging them), but a “random” set of Penrose tiles will almost certainly contain irregular shapes. A (new) aperiodic tiling with a very regular shape of each tile is no trivial thing.

Ī̲ stumbled, by chance, on an aperiodic tiling locally derivable from the Penrose tiling. Its set of tiles consists of three regular shapes: (72° rhombus), ⧫(36° rhombus), and ⬠(the regular pentagon); all having the same side length.

With the green curve commonly depicted on Penrose tilings, SVG format.
at Wikimedia Commons: More tiles, edges only; Both green and red curves.

Surely a set of rules can be formulated to ensure that only aperiodic tilings can be constructed from  ⧫ ⬠. One can notice that is employed in two distinct rôles; even counting this, we have only four sorts of tiles compared to 6 sorts of tiles (of 4 distinct shapes) from the original (but now almost forgotten) P1. By the way, it seems to me that the  ⧫ ⬠ tiling might be related to P1 closer than to P2 and P3.

It it really a new discovery worth publishing ? Should Ī̲ now finish the job by proving, from appropriate rules (see below), local derivability: Penrose tiling ⇐  ⧫ ⬠? If Ī̲ should, then give me, please, references to terminology and theoretical background that are currently in use by experts.

Update July 20: A sequence of simple rules is found that makes ♢ ⧫ ⬠ from a P2.
1. For each long side of a kite shared with another kite, dissect a 36°-72°-72° triangle (taking entire kite’s short side and smaller part of the long side with it) from the kite. After dissecting one triangle a trapezoid remains, and in the case of dissecting two the remnant is a ♢.
2. Unify each pair of these triangles sharing their short side to a ⧫.
3. Dissect each dart to two mirror-symmetric 36°-108°-36° triangles.
4. Unify each half-dart with the shape across its long side. This means, just erase all respective edges.

Update July 10: Now Ī̲ try to formalize rules that a  ⧫ ⬠ tiling derivable from Penrose tiling must obey. The conditions listed below are necessary, but Ī̲’m unsure neither about sufficiency nor about minimality.

1. Any ⧫ ⧫ pair may not share an edge.
2. Each ⬠ has two non-joint sides which it may share with other ⬠s only. No other ⬠’s side may be shared with another ⬠.
3. For each  one of its acute vertices is specified as inner and another as outer, with respective distinction for sides. Any ’s outer side may be shared with a ⧫ only. Any ’s inner side may not be shared with a ⧫.
4. When ♢ ♢ share an edge, their inner vertex must coincide. In other words, ♢ ♢ must join 72°-to-72°, 108°-to-108°, or may not form a parallelogram in their union (72°-to-108°).

Some fifth rule must stand to exclude certain “incorrect” vertex figures. My preferred formulation is:
5. when ♢ ⧫ share an edge, the ♢’s outer vertex must coincide with a ⧫’s acute (36°) vertex. In other words, ♢ ⧫ must join 72°-to-36°, 108°-to-144°, that is, may not join 72°-to-144°, 108°-to-36°.

Ī̲ have other variants for fifth rule, but this one has an advantage to be edge-based (as is customary in tiling theory) and pretty in line with the rule 4.

• It may have been used in Muslim architecture already. Marcus du Sautoy in his book "symmetry" describes the different kinds of symmetry used in Alhambra in Granada, he may be able to tell you more. – Sylvain JULIEN Jul 9 '17 at 15:12
• @Sylvain JULIEN: They certainly knew decagonal patterns since medieval times, but Ī̲ never saw such long (really endless) chains of ⬠s in Islamic art. – Incnis Mrsi Jul 9 '17 at 15:15
• In any case it is worth adding to the Wikipedia List of aperiodic sets of tiles – მამუკა ჯიბლაძე Jul 9 '17 at 16:27
• I don't see it listed at tilings.math.uni-bielefeld.de/glossary/mld-class-penrose either. If you want to know whether it's truly new, I'd suggest sending an email to an expert on tilings, perhaps Professor Goodman-Strauss: comp.uark.edu/~strauss. – mkreisel Jul 18 '17 at 0:46
• What is this strange thing you are doing with the capital version of the letter "i"? – Gerry Myerson Jul 18 '17 at 12:11

Seems like the following (not entirely rigorous, I admit) rules prove this tiling to be MLD with the Penrose P2 (darts-and-kites) tiling:

middles of darts $\iff$ blue edges

middles of kites $\iff$ those red edges which border two pentagons

In fact, as suggested by OP comments, the two tilings can be made closer to each other by switching to the next P2 generation. The rules however become slightly less obvious. Edges of the new P2 (yellow ones in the pictures below) are in 1-1 correspondence with the union of:

(1) all blue rhombi (each is cut symmetrically into two congruent acute triangles by a unique edge) and

(2) all pentagons (each is cut into an obtuse triangle and a trapezoid by a unique edge).

(doubleclick to enlarge).

As for decorations for kites and darts, all darts are again uniquely decorated, , but there are now three kinds of kite decorations -

"left", ,

"middle", ($\texttt{<->}$ 1-1 with blue edges of the OP tiling),

and "right", .

• The picture shows P2 ⇒♢ ⧫ ⬠ graphically. Nice to see decomposition explicitly, but Ī̲ evidently knew that after two or three inflations of P2 higher-generation kites and darts would each decompose to several whole or half-tiles – otherwise my generating code just wouldn’t show the picture we see! You said something about MLD, but did you specify packing rules for ♢ ⧫ ⬠? How many tiles do we distinguish? Do we count for orientation, and if we do, then shouldn’t we provide for two chiral ⬠s (note red/green edges)? These are details and nuances that prompted my question (i.e. that Ī̲ don’t know). – Incnis Mrsi Jul 9 '17 at 19:07
• Mmm I can only say that (clearly) each kite is decomposed in the same exact way, as well as each dart; and in the opposite direction, I think the rules I wrote determine kites and darts uniquely. What little I know about MLD is that one has to have explicit algorithms to switch back and forth, and I believe these are explicit algorithms, no? – მამუკა ჯიბლაძე Jul 9 '17 at 22:23
• Concerning what you call chiral ⬠s - I think it is just another way to formulate the same rules: darts correspond 1-1 to pairs of pentagons with blue edge in common (contain exactly one such pair symmetrically); and kites correspond 1-1 to pairs of pentagons with red edge in common (contain exactly one such pair symmetrically). – მამუკა ჯიბლაძე Jul 9 '17 at 22:28
• It seems that you don’t hear me. The letter “M” stands for mutual derivability. We both know how a ♢ ⧫ ⬠ tiling derived from a Penrose tiling is arranged, and it’s not a big deal to formulate rules to translate it back to Penrose, to P2 or else, to the generation of green curves or another (because inflation/deflation is an MLD too). This is a generally known stuff. On the other hand, when Ī̲ first posted the question, rules for ♢ ⧫ ⬠ (that are necessary for derivation of a Penrose) were completely missing. Here is the real problem. How the answer helped (or helps) to collect them? – Incnis Mrsi Jul 10 '17 at 6:50
• @IncnisMrsi, I hear you. I certainly did not formulate any explicit rules for your tiling. What I said is that given decorations of a kite and a dart it must be easy to reformulate the matching rules for P2 into those for your tiling. – მამუკა ჯიბლაძე Jul 10 '17 at 6:58

The system of ♢ ⧫ ⬠ prototiles might deserve research for its own merits, but (with derivation rules given) it isn’t very convenient for encoding Penrose tilings. Conditions sufficient for Penrose-equivalence do not formalize as matching rules for it. They can be made a sort of local conditions on blocks of tiles, but these are not rules about edge-sharing of the type “you may connect this side to that side with specified orientation”, nor do restrictions on vertex figures help with it.

As for “fourth”, “fifth” etc. set of rules for Penrose tilings, Ī̲ found other (albeit closely related) system of three prototiles much more convenient for derivation of them and serves as a transition from ♢ ⧫ ⬠ to the P3 system, and (by August7) yet another tiling, purely ♢ ⧫ ⬠ but looking differently, where (sufficiently) Penrosean matching rules exist.

For details, see the article Ī̲ wrote (but having difficulties to publish it even at arXiv.org due to their endorsement policy; see inside).