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
Upon an observation on David Smith’s Hedraweb blog, concerning Cairo* and Prismatic tiles and tilings thereof, various (as he terms it) ‘clusters’ of a broad basic set of 9 are initially formed (redrawn below), all convex hexagons, in various combinations and distributions, upon which the themed tilings are predicated. Upon initially trying to do the same myself, the study segued into a different, albeit related study. Simply stated, using the given clusters, I formed them into a hexagonal framework to see if a new tiling would be formed. Typically, regular hexagonal ‘holes’ would be formed on different scales. In short, tilings of Cairo, Prismatic and regular (oversized) hexagons were formed. Although ‘pleasing’, there is nothing particularly outstanding of note. However, upon the realisation that the hexagons permit subdivision into a series of basic polygons (which is well known), including Cairo and Prismatic tiles as well as other basic polygons (as above), the original tilings are transformed into a much more elegant, multi-combinatorial tiling that is particularly aesthetic. As a rule, the more the tiles are of a roughly equal area, the better, in that the tiling is better ‘balanced’, as against noticeably small and large tiles. As the subdivision causes tiles of like sizes, various harmonious overlays can be seen. For a given framework, there are multiple subdivisions. The tiles are colour-coded, with the same colour retained for a tile throughout.
* The dual model, although commonly given as the Cairo tiling, is not that as seen on the in situ pavings in Cairo, as I have discussed on my Cairo tiling page. (Nor is it the equilateral, as is also commonly given!)

PRESENTATION
The presentation is predicated on the 'Smith Set of 9'*, numbered (by myself) from left to right, with SMITH 1, SMITH 2, etc. For any one Smith block I show all possible distinct joinings in sections. For instance, SMITH 1 has a Set of 7 block. These are then in turn used to form tilings immediately below. First I show the (hexagonal) framework, then the (hexagonal) framework with a subdivided tiling of the 'holes'. It can be seen that for any one framework, it can be subdivided in a multitude of ways. A selection from my studies is shown, concentrating on the more aesthetic examples.
* Note that this is not a distinct set; Nos. 2 and 6 are reflections. However, for reasons of comparisons, I retain the Smith numbering as I see it.

DAVID GEORGES EMMERICH
As alluded to above, the study was originally of the Cairo and Prismatic tiles. However, in the course of my studies of trying to form new configurations with these tiles, the study expanded to include other tilings that formed as a by-product of my investigations. These include other ‘everyday’ polygons such as squares and regular hexagons. However, this is not an entirely new idea. David Georges Emmerich (1925–1996), a French architect, amongst his morphological studies also undertook some interesting work in tiling, with a series of 56 hand-drawn plates of tilings within a circular outline, placed on the FRAC site*, a French repository of contemporary art. Unfortunately, the date is (annoyingly) not given. Further annoyingly (for unclear reasons), these are shown at low resolution. Although still viewable, it is a trial looking at the diagrams and reading his (small text) classifications. The premise, overall, is unclear. Explanations are simply not given. What exactly is original or otherwise is not made clear. 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 obfuscations! Be that as it may, I am convinced that he has some original ideas, with many of the tilings bearing similarities to those shown here. However, the method he used to compose these is not given or cannot be inferred. A typical instance of the plates is shown below, with most of the tilings having relevance to the present study. The general murkiness is typical of all the plates.

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

Of Emmerich’s study, many interesting instances of Cairo, prismatic and other simple polygons in combination can be seen. However, these are scattered throughout his studies, and so are not particularly easy to analyse. Further, the tilings are shown as wireframes, resulting in the tiles not being easily distinguishable against a murky background. It really is frustration personified!

When I was composing my own tilings (2022), Emmerich’s work must have been in my mind, but 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 of the Centre-Val de Loire region in France, part of the national Frac network. It holds 15,000 drawings, 800 models, and 600 artworks, focusing on experimental architecture. It explores connections between art, architecture, and design through exhibitions and programming. It is based in Orléans. In 2013, it moved onto the site of a former military base, with a new museum building designed by Jakob + MacFarlane.

BAILEY STUDIES

SMITH 1

SMITH 1, BLOCK B TILINGS

The above is particulary pleasing, in that it consists of four basic pentagons; Cairo, Prismatic, Floret, and Hokusai/Moore! 

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