Cairo, Prismatic...

Or
Cairo (dual model*), Prismatic tiles in tiling combinations with other basic polygons, including equilateral triangles, squares, regular hexagons, kites, rhombuses, Hokusai/Moore pentagons, florets, all in various combinations.

INTRODUCTION
Following an observation on David Smith’s Hedraweb blog concerning Cairo* and Prismatic tiles, and the tilings derived from them, various “clusters” (as he terms them) from a broad basic set of nine are initially formed (redrawn below). These are all convex hexagons, arranged in different combinations and distributions, upon which the themed tilings are based.

When I first attempted to do the same myself, the study quickly segued into a different—though related—line of investigation. Simply stated, using the given clusters, I arranged them into a hexagonal framework to see whether a new tiling would emerge. Typically, regular hexagonal “holes” appeared at different scales. In short, tilings of Cairo, Prismatic, and regular (oversized) hexagons were produced. Although pleasing, there was nothing particularly remarkable about them.

However, once I realised that these hexagons permit subdivision into a series of basic polygons (a well‑known fact), including Cairo and Prismatic tiles as well as other fundamental polygons (as above), the original tilings transformed into a far more elegant, multi‑combinatorial arrangement with considerable aesthetic appeal. As a rule, the closer the tiles are in area, the better balanced the tiling appears, avoiding the visual imbalance of noticeably small and large tiles. Because the subdivision produces tiles of comparable sizes, various harmonious overlays become visible. For any given framework, multiple subdivisions are possible. The tiles are colour‑coded, with each tile retaining its colour throughout.

* The dual model, although commonly presented as the Cairo tiling, is not the same as that seen on the in situ pavings in Cairo, as I have discussed on my Cairo tiling page. (Nor is it the equilateral form, as is also commonly claimed.) 

PRESENTATION
The presentation is based on the ‘Smith Set of 9’*, numbered (by myself) from left to right as SMITH 1, SMITH 2, and so on. For each Smith block, I show all possible distinct joinings in separate sections. For example, SMITH 1 has a set of seven blocks. These are then used to form the tilings shown immediately below.

First, I present the hexagonal framework; then, the same framework with a subdivided tiling of the “holes.” It can be seen that any given framework can be subdivided in a multitude of ways. A selection from my studies is shown here, focusing on the more aesthetically pleasing examples.

* Note that this is not a strictly distinct set: Nos. 2 and 6 are reflections. However, for purposes of comparison, I retain the Smith numbering as I interpret it.

DAVID GEORGES EMMERICH
As alluded to above, the study was originally concerned with the Cairo and Prismatic tiles. However, in the course of attempting to form new configurations with these tiles, the investigation broadened to include other tilings that emerged as by‑products of the work. These include other “everyday” polygons such as squares and regular hexagons. This, however, is not an entirely new idea.

David Georges Emmerich (1925–1996), a French architect, undertook some intriguing work in tiling as part of his wider morphological studies. His output includes a series of 56 hand‑drawn plates of tilings within a circular outline, hosted on the FRAC site*, a French repository of contemporary art. Unfortunately, the date of these plates is (annoyingly) not given. Further—also annoyingly, and for unclear reasons—they are presented only in low resolution. Although still viewable, it is a trial to examine the diagrams and decipher his small, handwritten classifications.

The overall premise is unclear. Explanations are simply not provided, and what is original or otherwise is not made explicit. Among the plates are Penrose tilings and Voderburg spirals, so it is plainly not all original. Indeed, both Emmerich and FRAC seem to delight in obfuscation. Be that as it may, I am convinced that he does present some original ideas, and many of the tilings bear similarities to those shown here. However, the method by which he composed them is not stated and cannot be inferred. A typical example of the plates is shown below, with most of the tilings having relevance to the present study. The general murkiness is characteristic of all the plates.

Although Emmerich is published, it is in connection with his morphological studies rather than these tiling plates. This work is so little known that I have yet to see it cited in the tiling literature. He deserves to be better known.

Within Emmerich’s study, many interesting combinations of Cairo, Prismatic, and other simple polygons can be found. However, these are scattered throughout his plates and are therefore not particularly easy to analyse. Furthermore, the tilings are shown as wireframes, making the tiles difficult to distinguish against the murky background. It really is frustration personified.

When I was composing my own tilings (2022), Emmerich’s work must have been in my mind, though to what extent I do not recall.

*The Frac Centre–Val de Loire, formerly known as Frac Centre, is a public collection of contemporary art in the Centre–Val de Loire region of France, part of the national Frac network. It holds 15,000 drawings, 800 models, and 600 artworks, with a focus on experimental architecture. It explores connections between art, architecture, and design through exhibitions and programming. It is based in Orléans. 

BAILEY STUDIES

SMITH 1

SMITH 1, BLOCK B TILINGS

The above is particularly pleasing, as it comprises four fundamental pentagons: Cairo, Prismatic, Floret, and Hokusai/Moore. I have assigned the name of the last, drawing on the two earliest known instances: one by the famous Japanese artist Hokusai in 1824 and another by Herbert C. Moore in a 1909 patent. 

SMITH 2

Blocks B and C are omitted, as they repeat the tiling above

SMITH 3

SMITH 4

SMITH 5

SMITH 6 IS OMITTED,  AS IT REPEATS WITH BLOCK 2

SMITH 7 (PENDING RESSASSESSMENT)

SMITH 8 (PENDING RESSASSESSMENT)

SMITH 9 (PENDING RESSASSESSMENT)

OTHER COMBINATIONS

It will be seen that the compilations above are all of the same block. However, this is not the only possibility! An obvious thought is to combine two different blocks. The possibilities are immense, so much so that  I only looked at a sample, as below, to establish the principle. Simply stated, the study was becoming overwhelming, hence the short look.  Below I show two tiles I term as 'shields'.

CAIRO-PRISMATIC TILING
In related matters are references to the tilings forming the bedrock of the study, namely the Cairo-Prismatic tilings. The study of these is a relatively recent introduction, seemingly introduced as a term by Frank Morgan, in 2012, in regard to perimeter-minimising pentagonal tilings. Subsequently, there has been little interest, with some of the papers below overlapping; a case in point is Ping Ngai Chung. Marjorie Rice has the first known drawing, in 1981.
No claim is made for completeness here, of an initial survey of the literature.

References

Berry, John, Matthew Dannenberg, Jason Liang, Yingyi Zeng. ‘Symmetries of Cairo-Prismatic Tilings’. Rose-Hulman Undergraduate Mathematics Journal, Volume 17, No. 2, Fall, 2016, pp. 40–60.

The paper was apparently an offshoot of Frank Morgan and Williams College (Berry was a student). Amazingly, none of the above are known for their tiling interest! I am very much impressed by this paper.

https://scholar.rose-hulman.edu/rhumj/vol17/iss2/3/

Chung, Ping Ngai, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, Elena Wikner. ‘Isoperimetric Pentagonal Tilings’. Preprint, August 11, 2011, pp. 1–16.

This is the forerunner to the 2012 paper in AMS Notices. Although the substance is the same, there are differences, noticeably so in places.

All the people above were students of Frank Morgan.

https://arxiv.org/abs/1111.6161

Chung, Ping Ngai, Miguel A. Fernandez, Niralee Shah, Luis Sordo Vieira and Elena Wikner. ‘Perimeter-minimizing pentagonal tilings’. Preprint 2011 (not seen).

This is the forerunner to the 2014 paper in involve.

Chung, Ping Ngai, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, Elena Wikner. ‘Isoperimetric Pentagonal Tilings’. Notices of the American Mathematical Society 59:5, 2012, pp. 632–640.

This has much in common with the following article by Chung et al as well as many of the same authors. Cairo-Prismatic tilings are implied by the title.

Named tilings include: Spaceship, Pills, Stripes, Sardines, Bunny, Plaza, Christmas Tree, Windmill, Chaos, Waterwheel, and Rice. No personal credit is given to any named tiling.

https://www.ams.org/journals/notices/201205/rtx120500632p.pdf

Chung, Ping Ngai, Miguel A. Fernandez, Niralee Shah, Luis Sordo Vieira and Elena Wikner. ‘Perimeter-minimizing pentagonal tilings’. involve [sic] 2014, Vol. 7, No. 4.

Note also the earlier publication by Chung et al, with many of the same authors; Fernandez, Viera, Wilkner (Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata are omitted).

Of note is a different interpretation of the Cairo-Prismatic tilings, with its dual.

Named tilings include: Spaceship, Pills, Stripes, Sardines, Bunny, Plaza, Christmas Tree, Windmill, Chaos, and Waterwheel. No personal credit is given to any named tiling. Three tilings are unnamed.

The second half of the paper has an extensive study on tiling tori (way beyond my understanding)

https://projecteuclid.org/journals/involve-a-journal-of-mathematics/volume-7/issue-4/Perimeter-minimizing-pentagonal-tilings/10.2140/involve.2014.7.453.pdf

Emmerich, David Georges. Frac Centre
https://collection.frac-centre.fr/artwork/david-georges-emmerich-tessellations-composites-5030000000004824?page=1&filters=query%3Aemmerich&note

All 56 plates. Cairo-Prismatic and other combination tilings appear intermitently throughout.

'Mathgrrl'. 'Cairo and Prismatic Pentagons'. MakerHome Blog. May 18, 2014.

Based on Frank Morgan's work.
https://makerhome.blogspot.com/2014/05/day-265-cairo-and-prismatic-pentagons.html

Morgan, Frank. 'Bubbles and Tilings: Art and Mathematics'. Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture, pp. 11–18. 

Pp. 14–18 discuss Cairo-Prismatic tilings.

https://archive.bridgesmathart.org/2014/bridges2014-11.pdf

Morgan, Frank. ‘New Optimal Pentagonal Tilings’. First published 27 May 2014, with updates 31 January 2015, 11 February 2015 and 3 April 2019.

https://sites.williams.edu/Morgan/2015/01/31/new-optimal-pentagonal-tilings/

Morgan, Frank. Frank Morgan - Optimal Pentagonal Tilings - CoM May 2021

https://www.youtube.com/watch?v=PpUx0nnWfKQ&t=592s

A good explanation, including the people involved.

Schattschneider, Doris. In ‘In Praise of Amateurs’, pp. 140–166. In The Mathematical Gardner. David A. Klarner, editor, Wadsworth Inc., 1981.

A collection of articles in honour of Martin Gardner, with tiling featuring prominently. Majorie Rice features prominently, albeit only pp. 162–163 concerns the Cairo-Prismatic tiles, but is not titled as such, and is only captioned and not discussed. Seemingly, it was named ‘Cairo-Prismatic’ by Morgan et al, as late as 2012.

Smith, David. 'Cairo-Prismatic Tilings- Part One'. Hedraweb blog, 6 November 2018.
https://hedraweb.wordpress.com/2018/11/06/cairo-prismatic-tilings-part-one/

Seven tilings.

Smith, David. 'Cairo-Prismatic Tilings-Part Two'. Hedraweb blog, 7 November 2018.

https://hedraweb.wordpress.com/2018/11/07/cairo-prismatic-tiling-part-two/

Six tilings. The last tiling alludes to the possibilty of a 'regular hexagon void at its centre', but is not shown.

Stewart, Ian. ‘The Art of Elegant Tiling’. Scientific American, July 1999, pp. 96–98.

Shows a tiling by Rosemary Grazebrook of Cairo tiles (with occasional subdivisions) and regular hexagons, but no Prismatic tiles, p. 97. 

Page Created 18 December 2025. Last Updated 6 January 2026