Mathematical shapes and intellectual property – a legal perspective on the "einstein tile"

Former Guest Kat Peter Ling brings us a fascinating discussion about the overlap of two dimensional geometric planes and intellectual property.

Sadly, this former GuestKat lacks absolutely basic DIY skills to enable him to tile his own bathroom floor. He has nonetheless been long fascinated by various unresolved mathematical problems, among which are the different ways to tile a two-dimensional plane.

The most famous such problem (arguably) is the so-called "einstein problem", for which a group of mathematicians recently proposed a solution that looks stunningly elegant. After describing the problem and its possible solution (the publication has not yet been peer-reviewed), I will examine potential legal intricacies related to the protection of such shapes in intellectual property law.

                                                                                   Looking for the geek
  
The "einstein problem"

Geometrically speaking, a two-dimensional plane can be filled with tiles of infinitely many different shapes and sizes. However, both in mathematics and home improvement, the only solutions that are relevant are those where the tiles fill the plane without gaps or overlaps.

Think trying to tile a floor with neat square-shaped tiles, as compared with not-so-neat circle-shaped ones. The first yields both a valid geometric solution and a nice-looking floor in a bathroom, the second does neither.

Both in geometry and home improvement, square tiles may be a working solution, but they are also boring. Wherever you look, there is the same, ever-repeating pattern of squares. Is it possible to create a pattern that is "aperiodic", i.e. never repeats itself? And, importantly, is it possible to use one single shape of tile to create such a pattern?

In geometry, this question is referred to as the "einstein problem" (a pun based on the German "ein Stein", i.e., "one stone", and not the name of the Nobel Laureate physicist, hence the lowercase "e"). Wikipedia defines the problem as the search for "a single prototile that by itself forms an aperiodic set of prototiles; that is, a shape that can tessellate space but only in a nonperiodic way".

Such a pattern would "walk a fine line between order and disorder, admitting tilings, but only those without any translational symmetry, never permitting the simple repetition of periodic tiling" (Smith et al. (2023)).

In 1974, British Nobel Laureate Sir Roger Penrose proposed solving the problem using two different tile shapes and proved that the resulting pattern is aperiodic. This pattern became known as the "Penrose tiling".

                                                                                      Penrose tiling

In 2023, David Smith, a retired print technician and recreational mathematician, announced that he has found a solution to the problem using only one stone that is "almost mundane in its simplicity", namely a 13-sided polygon that he dubbed the "hat":

                                                                                            The "hat"

This pattern required the use of both a "reflected and an "unreflected" version of the "hat" (in other words the "hat" and the mirror image of the "hat"), making some doubt whether this was the true "einstein tile". So, three weeks ago, the same authors published a further development of their work, a "chiral aperiodic monotile" that tiles the plane alone, without need for its mirror image, dubbed the "spectre" or the "vampire einstein" (as the shape is "not accompanied by its reflection"):

                                                                                      The "spectre"

I will look at whether, and if so how, the shape of the "einstein tile", or patterns of "einstein tiles", can be protected under intellectual property law.

How the "einstein tile" might be protected by intellectual property law

Patents are granted for new inventions applicable in industry. The einstein tile (in its purest mathematical form) is an abstract solution of an abstract problem, i.e, not applicable in industry. Sir Roger Penrose (who came up with the two-tile solution to the same problem, see above) solved this issue by claiming patent protection for a "set of tiles for covering a plane surface" (US 4,133,152).

Claiming actual tiles for tiling an actual floor indeed makes the abstract idea industrially applicable and thus patentable.

However, is the solution to a mathematical problem even an "invention"? Was the einstein tile really invented, rather than merely discovered by Smith et al.? This question is obviously bigger than the einstein problem itself, going to the philosophical fundaments of mathematics.

Nonetheless, to establish patentability, it may be helpful to remember that the authors did not only identify the "spectre", but proved that an infinite set of aperiodic polykites exists (i.e., an infinite number of different shapes that all tile the plane aperiodically). As such, the "spectre" is the smallest (and arguably most elegant) member of the set.

Using this shape to tile a floor (rather than any of the infinitely many other possible shapes they found), e.g., because it is the most pleasant to look at or the easiest to cut, can help establish that the shape of the "spectre" is particularly well-suited to solve the technical problem and thus that it is an invention.

(By the way, according to online sources (here and here), Sir Roger Penrose actually sought to enforce his patent against Kimberly-Clark in the late 1990s for using the "Penrose Tiling" imprinted on toilet paper. The parties are reported to have settled early, so little is known about the case.)

The "spectre" would be an obvious candidate for design protection. A design is the "appearance of the whole or a part of a product resulting from the features of, in particular, the lines, contours […]" (Art. 3(a) Community Design Regulation). However, a shape "solely dictated by its technical function" is excluded from design protection and the shape of the "spectre" is dictated exclusively by its technical function.

Does it help that many other shapes exist that do the same thing? In C-395/16 Doceram, the Court of Justice argued that the "technical function" test must not take into account the existence of alternative designs.

If the claimed shape is technically necessary, it is excluded from design protection even if other shapes exist that fulfil the same technical effect. In other words, the infinite set of other shapes that tile the plane aperiodically will not help in obtaining design protection for the "spectre".

On a side note, I do not believe that even a pattern of "spectres" (such as on the above picture) can be registered as a design. Design law requires that the applicant file a "representation of the design" with the register (e.g. Art. 36(1)(c) CDR). Given that a pattern of "spectres" is aperiodic, mathematically speaking it is not possible to file a "representation" of the pattern: every instance of it is unique.

What about trade marks? If the shape claimed protection for "tiles" in class 19, it might be objected to for being "the shape […] of goods which is necessary to obtain a technical result" (Art. 7(1)(e)(ii) EUTMR).

The CJEU looked into the relationship between trade marks and mathematical shapes in the Gömböc case (C-237/19). Like the "einstein tile", the Gömböc is the solution to a mathematical problem. The Gömböc is a three-dimensional body with only two equilibrium points, an object that was conjectured not to exist – before two mathematicians proved the conjecture wrong and built the 3D-object.

Since the Gömböc's shape defines its function (the object returns to its stable equilibrium point despite the manner that it is placed on a plane), the shape is necessary to obtain a technical result. The CJEU accordingly concluded that it is not registrable as a trade mark (see Katposts here and here).

If the above-mentioned Doceram case can be deemed to also apply to trade marks, I am afraid the CJEU in Gömböc has doomed trade mark protection for the "spectre" in the EU.

Finally, can the "spectre" constitute a work protected under copyright law? Copyright protects works that are original in the sense that they are the author's own intellectual creations (CJEU, C-5/08 Infopaq). In EU law, a work is original if—

"it reflects the personality of its author, as an expression of his free and creative choices.... [W]hen the realisation of a subject matter has been dictated by technical considerations, rules or other constraints, which have left no room for creative freedom, that subject matter cannot be regarded as possessing the originality required for it to constitute a work" (CJEU, C‑683/17 Cofemel).

Finding the solution to a mathematical problem is always "dictated by technical considerations, rules or other constraints" and involves no creative choices. There is no space to exercise a "creative ability in an original manner by making free and creative choices" (CJEU C-833/18 Brompton Bicycle).

In my view, this goes not only for the "spectre", but also for all other members of the infinite set of aperiodic tiles found by Smith et al. Hence, the "spectre" or patterns of "spectres" are unlikely to be protected by copyright.

Conclusion

It seems fitting that the most promising way to protect a mathematical result is primarily through its industrial application (patents) and not through its appearance. Nonetheless, the above is just an initial analysis and more can possibly be said on (creative) strategies to protect the "einstein tile", in particular, as a design or even a trade mark.

Actual "einstein tiles" may soon pop up in hardware stores, and bathrooms of the future might be tiled with more interesting patterns than ever before. As Einstein (this time the capital E Nobel Laureate), once wondered, how can it be "that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?"