Listing All Instances of the Cairo
Tiling in Print
The
criteria for the listing is of the Cairo tiling in text and/or in diagrammatic
form, in which I list all instances of the Cairo tiling, as of the ‘standard
model’ whose parameters can vary with basic features of a Cairo tiling, i.e. a
symmetrical pentagon (not necessarily of equal lengths), with two opposite right angles. The list is shown in different
ways, as according to various filters:
- A simple
listing of attributed references i.e. mentioned in association with
Cairo, e.g. 'is a favourite street-tiling in Cairo', arranged as according to chronology, with
referral to the quote/diagram in question, in effect a simple
bibliographic listing.
- References
of all instances of the Cairo tiling as a
pentagon per se, of a 'standard model', Cairo attributed or not, with quotes and comments thereof where
appropriate.
Due
to the dual nature, there are on occasions overlaps in the text. Can anyone add
to this list, with either other references (the earlier the better) or of any
additional further information?
1. A simple
listing of attributed references, mentioned
in association with Cairo, e.g. 'is a
favourite street-tiling in Cairo' arranged as
according to chronology, with Cairo quote
1.
1971 Dunn, J. A. ‘Tessellations with Pentagons’. The
Mathematical Gazette, Vol. 55, No. 394 December, pp. 366-369
…and is a favourite street-tiling in Cairo.
A first-hand sighting, one of only two; also
see Macmillan.
2.
1975 Gardner,
M. Scientific American. Mathematical Games, July. ‘On tessellating the
plane with convex polygon tiles’, pp.112-117 (p. 114 and 116 re Cairo aspect)
This beautiful tessellation is frequently seen as a street tiling in Cairo,
and occasionally on in the mosaics of Moorish buildings.
The
second recorded attribution. Gardner’s
account is interesting, in different ways. As such, this is not a first-hand
sighting; although not stated as such, upon research this is based on James A. Dunn’s paper. The ‘Moorish buildings’ aspect is a reference to Richard K. Guy’s
account of a supposed sighting at the Taj Mahal, almost certainly a mistaken account
on his part (I asked him).
.
3. 1978 Schattschneider, D. Mathematics Magazine, January.
‘Tiling the Plane with Congruent Pentagons’, pp. 29-44 (p. 30 re Cairo
aspect).
It is said to appear as a street paving in Cairo
Likely referring to Martin Gardner or James Dunn’s quote; both authors
are mentioned in the bibliography.
4.
1979 Macmillan, R.H. Mathematical Gazette, 1979.
‘Pyramids and Pavements: some thoughts from Cairo’,
pp. 251-255.
Many of the streets of Cairo
are paved with a traditional Islamic tessellation of
pentagonal tiles, as shown in Fig. 4. The pentagons are all identical in size
and shape, having four sides equal and two of their angles 90°…
A firsthand sighting, the second
of only two, and so of the utmost significance; also see Dunn. No reference is
made in the article itself or the references to any of the three above articles,
and so this is likely an independent account, as a ‘discovery’. As such, this
is a little surprising, in that Dunn’s article was also from the Mathematical
Gazette! Of note is the reference to the tiles being coloured, or arranged
of the same colour, ‘back to back’, this being the first recorded instance;
indeed, the only one!
5.
1982 Martin, George E. Transformation Geometry: An
Introduction to Symmetry, p. 119.
The beautiful Cairo
tessellation with a convex equilateral pentagon as its prototile is illustrated
in Fig. 12.3. The tessellation is so named because such tiles were used for
many streets in Cairo.
Likely referring to Martin Gardner or James Dunn’s quote.
6. 1984 Blackwell, William. Geometry in Architecture,
p. 54, Wiley 1984.
This unusual pattern, which is seen in street tiling in Cairo
and occasionally in the mosaic of Moorish buildings
Likely taking from the Gardner
quote, as the latter part is almost word for word.
7.
1986 McGregor, James and Watt, J. The Art of
Graphics for the IBM PC, p. 196-197.
…is the well-known Cairo
tile, so called because many of the streets of Cairo
were paved in this pattern
8.
1986 Ehud, Bar-On. Computers and Education.
Elsevier
… especially the Cairo tiling.
9.
1989 Chorbachi, W. K. Computers and Mathematics with
Applications. ‘In the Tower
of Babel:
Beyond Symmetry In Islamic Design’. Vol. 17, No. 4-6, pp. 751-789 (Cairo
aspects 783-794)
The pattern of a favorite street tiling in Cairo
Likely quoting from Dunn, as
he is mentioned in the article (Note US spelling of favourite, note that
Chorbachi also omits the dash between ‘favourite’ and ‘street’.
10.
1989 Harpe, P. De La. Quelques Problèmes Non Résolus en
Géométrie Plane. L’Enseignement Mathématique, t 35 (1989), pp. 227-243 (in
French) Cairo tiling
on p. 232
…dans les rues du Caire (…on the
streets of Cairo)
Likely taken from George Martin, given that the (‘unusual’) configuration of
the diagram is the same.
11.
1990 Hill, Francis S. Computer Graphics.
Macmillan Publishing Company, New
York, p. 145.
An equilateral pentagon can tile the plane,
as shown in Figure 5.4. This is called a Cairo
tiling because many streets in Cairo
were paved with tiles using this pattern…
Likely quoting from McGregor and Watt, given that the text is very much
alike, and their work is quoted and illustrations are used in the book.
12.
1991 Fetter, Ann E et al. The Platonic Solids
Activity Book. Key Curriculum Press/Visual Geometry Project. Backline
Masters, pp. 21, 97
This pattern is seen in street tiling in Cairo
and in the mosaics of Moorish buildings
Likely referring to the Gardner
quote, both parts is almost word for word.
13.
1991 Wells, David. The Penguin Dictionary of Curious
and Interesting Geometry. Penguin Books, pp 23, 61, 177.
So called because it often appears in the streets of Cairo,
and in Islamic decoration.
Likely referring to the Gardner
quote, both parts are almost word for word.
14.
1994 Leathard, Audrey. Going inter-professional:
working together for health and welfare
In the Cairo
tessellation (Wells 1991)…
Quotes the Wells reference. A very minor account. Note that this
reference is only included for the sake ‘of everything’; the book is apparently
of a non-mathematical nature, and is not illustrated with the tiling.
15.
1994 Carter
Bays.
Complex Systems Publications, Volume
8, Issue 2, 127-150, Cairo
aspect p. 148
'Cellular Automata in the Triangular Tessellation’
… the Cairo tessellation (a tiling of identical equilateral pentagons)…
Cursory mention in passing
16. 1996
O’Keefe, Michael and Bruce G. Hyde. Crystal Structures No. 1. Patterns & Symmetry.
Mineralogical Society of America
p. 207
The
pattern is known as Cairo tiling, or MacMahon’s net
and In Cairo
(Egypt)
the tiling is common for paved sidewalks…
17.
1998 Singer, David A. Geometry Plane and Fancy,
1998, pp. 34 and 37.
One particularly elegant tiling of the plane by pentagons is known as the Cairo
tessellation, because it can be seen as a street tiling in Cairo….
18.
1997 Serra, Michael. Discovering Geometry: An Inductive Approach. Key
Curriculum Press, p. 404
The Cairo street
tiling shown at right is a very beautiful tessellation that uses equilateral pentagons
(the sides are congruent but not the angles).
Not seen, Google Books reference.
19. 2000 Deza, M. et al. Fullerenes
as tilings of surfaces. Journal of Chemical Information Computer and
Modelling. ACS Publications, pp. 550-558
…
is the Cairo tiling…
20.
2001 Edward Duffy, Greg Murty,
Lorraine
Mottershead. Connections Maths 7. Pascal
Press, p. 83
Cairo
streets have this Islamic pattern
21.
2003 Teacher’s Guide: Tessellations and Tile
Patterns, p.30 (Cabri) Geometric investigations on the VoyageTM 200 with
Cabri. Texas Instruments Incorporated
….Probably the most famous of these pentagonal patterns is the ‘Cairo
Tessellation’ named after the Islamic decorations found on the streets of Cairo…
Begins by quoting David Wells’ book … Curious… and likely the
text is based on his reference. However, the ‘Teacher’s Guide’ gives a
different tiling, interestingly a ‘collinear’
pentagon.
22.
2003 Gorini, Catherine A. The Facts on File Geometry
Handbook. 2003, 2009 revised edition. Facts on File Inc, and imprint of
Infobase publishing, p. 22.
Cairo tessellation: A tessellation of the
plane by congruent convex equilateral pentagons that have two nonadjacent right
angles; so called because it can be found on streets in Cairo.
Oddly, Gorini shows an accompanying picture of a pentagon that is not
equilateral, a 4, 1 type…
23. 2003 Pritchard, Chris. The
Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry
Teaching, P. 421-427, Cairo
aspect p. 421.
Is a favourite street tiling in Cairo
This is an anthology, and simply repeats Dunn’s article, and follow-up
correspondence. Nothing original is shown.
Not seen, Google Books reference.
24.
2004 Parviainan, Robert.
‘Connectivity Properties of Archimedean and Laves Lattices’. Uppsala
Dissertations in Mathematics 34. p. 9. 2004.
The lattice D (3^{2}.
4. 3. 4) is sometimes called the Cairo lattice, as the pattern occurs frequently as
tilings on the streets of Cairo.
25. 2005 Mitchell, David. Sticky
note origami: 25 designs to make at your desk, Sterling Publication
Company, p. 58-61.
The Cairo Tessellation is an attractive
and intriguing pattern of tiles named as a result of its frequent occurrence on
the streets of Cairo
and in other Islamic centers and sites
26.
2005 Phillips, George McArtney. Mathematics Is Not a Spectator Sport.
Springer, p. 193
This is called the Cairo
tessellation.
27.
2005 Wilder, Johnston-Sue; John Mason, Developing
Thinking in Geometry, p. 182.
… is often referred to as the Cairo
tessellation as it appears in a mosque there.
28. 2005
Bays, Carter. Complex Systems Publications,
Volume 15, Issue 3, 245-252, Cairo aspect p. 249-250
‘A Note on the Game of Life in Hexagonal and Pentagonal Tessellations’
‘Here we have chosen the Cairo tiling…’
29. 2006 Sharp, John. ‘Beyond Su Doku’.
Mathematics Teaching in the Middle Years.
Vol. 12, No. 3 October 2006 pp. 165-169 pp. 167-169, in the context of a ‘Cairo Su Doku’.
Cairo
tile So Doku with two overlapping hexagons
30. 2007 B. G. Thomas and M. A. Hann. In Sarhangi, Reza (Ed).
Bridges. Mathematical Connections in Art, Music, and Science. (Ninth)
Conference Proceedings 2007. Donostia,
Spain.
Patterned Polyhedra: Tiling the Platonic Solids
There are however various equilateral pentagons that can tessellate
the plane. Probably the best known is the Cairo tessellation…
** 2007 Ollerton, Mike.
100+ Ideas for Teaching Mathematics p.66
This
tessellation not only begs interesting questions about angle sizes and side ...
The Cairo tessellation... A The challenge is to use this tile to fill 2D space.
31.
2008 Anon. Key Curriculum Press. Chapter 7,
Transformations and Tessellations, p. 396.
The beautiful Cairo street
tiling shown below uses equilateral pentagons.
Does anyone know of this book? I found it as
a ‘part PDF’, without a title
32. 2008 Goos, Merrilyn et al. Teaching Secondary
School Mathematics: Research and Practice for the 21^{st} Century.
The particular tiling pattern of an irregular pentagon, shown in Figure
9.16, is called the Cairo
tessellation because it appears in a famous mosque in Cairo.
Not seen, Google Books reference. Very curious; the ‘famous mosque’ has evaded detection! Likely
quoting, and extrapolating, from Gardner.
33.
2008 Thomas, B. G. and M. A. Hann in Sarhangi, Reza
(Ed). Bridges. Mathematical Connections in Art, Music, and Science.
(Tenth) Conference Proceedings 2008. Leeuwarden,
Netherlands
p. 102 in ‘Patterning by Projection: Tiling the Dodecahedron and other Solids’
There are, however, equilateral convex pentagons that do tessellate
the plane, such as the well known Cairo
tessellation shown in Figure 1.
Also, other minor references essentially in passing.
34. 2008 Thomas, B. G. and M. A. Hann. In Sarhangi, Reza (Ed). Bridges.
Mathematical Connections in Art, Music, and Science, p. 101.
Probably the best known is the Cairo tessellation
35. 2008 Kaltenmorgen, Birgit. Der mathematische Patchworker. (in
German) Wagner, Gelnhausen; 1^{st} edition 2008, pp. 82-83
Fünfeck beim Cairo-Tiling
35.
2008 Fathauer, Robert. Designing and Drawing
Tessellations, 2008, p. 2.
A common street paving in Cairo,
Egypt
is shown above left.
36.
2009 Kaplan, Craig. S. Introductory Tiling Theory
for Computer Graphics. Morgan & Claypool Publishers, p. 33.
The Laves tiling [3^{2}. 4. 3. 4] is sometimes known as the ‘Cairo
tiling’ because it is widely used there.
37. 2010 Alsina, Claudi and Roger B. Nelsen. Charming Proofs: A
Journey Into Elegant Mathematics. Dolciani Mathematical Expositions,
p. 163.
This rather attractive monohedral pentagonal tiling is sometimes called the Cairo
tiling, for its reported use as a street paving design in that city.
38. 2010 Elwes, Richard. Maths
1001: Absolutely Everything You Need to Know about Mathematics in 1001
Bite-Sized Explanations. Quercus, p. 109
…it adorns the pavements of that city’ (Cairo).
39. 2010 Ressouche, E. et al. Magnetic frustration in an iron-based Cairo
pentagonal lattice. Physical Review Letters.
A famous one being the Cairo tessellation whose name was given because it
appears in the streets and in many Islamic decorations
40.
2011 Rojas, M. et al. Frustrated
Ising model on Cairo
pentagonal lattice
Cairo pentagonal lattice
41. 2011. Askew, Mike and Sheila
Ebbutt. The Bedside Book of Geometry: From Pythagoras to the
Space Race: The ABC of Geometry Murdoch
Books Pty Limited
42. 2011 Goldemberg,
Eric. Pulsation in Architecture p.338
Housing Exhibition in Vienna,
Austria Project Description The Cairo Pods gave SPAN ... The Cairo Tessellation,
known in mathematics also as an example of ...
43. 2012 Edited
by Christoph Gengnagel, A. Kilian, Norbert Palz, Fabian Scheurer. Computational Design
Modeling: Proceedings of the Design Modeling Symposium. Springer, p. 229...
…on the mathematical configuration of a Cairo tessellation
44. 2013 Iwamoto,
Lisa. Digital Fabrications: Architectural and Material Techniques
… project uses a pentagonal Cairo
tessellation pattern, flexibly aggregated to yield multiple overall
arrangements. Each vertical layer of the cell was ...
45. 2014 Gorski, Hans-Joachim
and Susanne Müller-Philipp. Leitfaden
Geometrie: Für Studierende der Lehrämter, Springer p. 186
2.
References of all instances of the Cairo tiling, attributed or not, in chronological
order, with quotes and comments where appropriate
1. 1909 Moore,
Herbert C. ‘Tile’. United States Patents 928,320 and 928,321 of 20 July 1909
The first recorded instance,
of a patent for a flooring. The tiling also appears in the second patent. As
far as I am aware, no one has quoted his work in the Cairo tiling. Does anyone
know of Moore at all? He is connected with Boston, Massachusetts, USA. Were the
tiles actually made? Of note here is that Moore in principle shows minimum and
maximum deformations of the pentagon, to a implied rectangle and square, as
outlined in more detail by Macmillan in his 1979 paper.
2. 1921 MacMahon, P. A. New Mathematical Pastimes. Cambridge
University
Press 1921 and 1930. (Reprinted by Tarquin Books 2004)
Cairo
diagram p. 101
3. 1922
MacMahon, P. A. ‘The design of repeating patterns for decorative work’. Journal of the Royal Society Arts
70 (1922), 567-578. Related discussion ibid pp. 578-582
Of note is that MacMahon refers to a ‘haystack’,
meaning a Cairo
tile, p. 573, after fig. 13. This term is also interestingly used by him in New
Mathematical Pastimes. His nephew W. P. D. MacMahon also uses this word
(haystack) in ‘The theory of closed repeating polygons…’, so confusion arises
as to whom exactly determined the tile.
4. 1925 Haag, F. 'Die pentagonale Anordung von sich berührenden Kriesen in der Ebene’. Zeitschrift fur Kristallographie 61 (1925), 339-340
Has Cairo tiling in the form
of circle packing
5. 1926 Haag, F. 'Die Symmetrie
verhältnisse einer regelmässigen Planteilung’. Zeitschrift für mathematischen und naturwissenschaftlichen Unter-richt,
Band 57 (1926), 262-263
Has Cairo tiling in the form of circle packing
6. 1931 Laves, F. ‘Ebenenteilung in
Wirkungsbereiche’. Zeitschrift für Kristallographie 76 (1931): 277-284.
7.
1933 Bradley, Amos Day. The
Geometry of Repeating Design and Geometry of Design for High Schools.
Bureau of Publications
Teachers College,
Columbia University,
New York
City, and 1972 reprint. Book as oft-quoted by Schattschneider, but surprisingly
no one else.
P.123 Cairo-like diagram, dual of the 3. 3. 4. 3. 4. Possibly based on the work of Haag, of which the diagram resembles, and of whom articles he quotes
8.
1951 Cundy H. Martyn; Rollett, A.P. Mathematical
Models. Oxford University Press (I have the second edition, of 1961).
‘We have space for one of his; [MacMahon’s] it consists of
equal-sided (but not regular) pentagons, but has the appearance of interlocking
hexagons (Fig. 58)’
Cairo
diagram (but not attributed) p. 63 (picture) and p. 65 (text). The diagram is
derived from MacMahon’s book, as Cundy freely credits.
9.
1951 'Croton'. Cairo
tiling used as a crossword puzzle, in The Listener, 13 December 1951,
puzzle 1128 Hexa-Pentagonal I, by 'Croton'.
'Croton' is a pseudo-name; it's somewhat of a long shot given the time passed
since the puzzle’s inception, but the does anyone know he is?
10.
1954 'Croton'. Cairo
tiling used again as a crossword puzzle, in The Listener, 22 April 1954,
puzzle 1251 Hexa-Pentagonal II, by 'Croton'.
11.
1955 Rollett, A. P. ‘A Pentagonal Tessellation’. The
Mathematical Gazette, Vol 39, No. 329 (Sep. 1955) note 2530 p. 209.
Rollett states 'My colleague Mr. R. C. Lyness noticed this [Cairo
tiling] pattern on the floor of a school in Germany.
It has also appeared in a crossword puzzle in The Listener'.
The detail given is infuriatingly sparse to try and locate this sighting. Does
anyone know about Lyness's connection to Germany,
and if so where is it? By 'school' does he mean university? 'The Listener'
reference has been found; see 1951 and 1954 entries above.
12. 1956
Langford, C. Dudley. ‘Correspondence’. The Mathematical Gazette, Vol.
40, No. 332 May 1956 p. 97
Drawing
readers' attention to MacMahon’s Cairo tiling picture in New
Mathematical Pastimes. Of importance, due to Cairo tiling reference,
referring to Rollett’s piece in the Gazette (Note 2530). Also of note in that
Langford gives a different construction to MacMahon’s. Also see T. Bakos, which
completes a non stated ‘trilogy’ of writings of the day.
13. 1958 Bakos, T. ‘2801 On Note
2530’ (Correspondence on C. Dudley
Langford's 'Cairo' tile reference)’. The Mathematical Gazette, Vol 42, No.
342 December 1958, p. 294
Of
importance, due to Cairo tiling reference,
referring to Rollett’s and Langford’s pieces in the Gazette (Note 2530 and
correspondence). Gives an interesting discussion in terms of minimum values of hexagon and pentagon.
14. 1963 Coxeter, H. S. M. Regular
Polytopes. Second edition. Dover Publications Inc., New
York.
Cairo
diagram (but not attributed) on the cover. Interestingly, this is likely the
first instance of using different coloured subsidiary hexagons to better
feature the overlapping hexagon aspect.
The type of pentagon is not clear due to the nature of the drawing, with
somewhat thick lines, but it would appear to be equilateral. An open question
is does this appear on (or in) the first edition of 1947? I have not got the
book to hand.
15. 1967
Wood, D. G. ‘Space Enclosure Systems’, Bulletin 203. Columbus,
Ohio:
Engineering Experiment Station, The Ohio
State
University.
PP. 3-4, 30-31
Wood (a professor of
industrial design rather than a mathematician) makes a curious observation as
regards tilings with equal length sides, with the later to be known Cairo
tiling being one of five such instances (the equilateral triangle, square, Cairo
pentagon, hexagon, rhomb); as such, I do not recall seeing this simple
observation elsewhere. Is this significant? Much of Wood’s work here, and
elsewhere in the book, is in regards to prisms, of which he shows a ‘Cairo’ prism. Does anyone know of Wood? At the time of
writing he would be 99. Is he still alive? Did he do anything further with the
tiling? He freely credits both MacMahon and Cundy and Rollett as the source of
the pentagon per se, the observation of his appears to be his own.
16.
1969? Critchlow, Keith. Order in Space. A Design
Source Book. Thames & Hudson. A date of 1969 is given in the book but
it is unclear if this was when it was first published. The published date is
apparently given as 1987. 2000 reprint.
Cairo
diagram (but not attributed) p. 49, but no text. This also has an interesting
series of diagrams p. 83, best described as ‘variations’ with Cairo-like
properties, with ‘par hexagon pentagons’ combined in tilings with regular
hexagons.
17.
1970 Ranucci, Ernest R. Tessellation and Dissection.
J. Weston Walch
Cairo-like
diagram (but not attributed) p. 36 (picture and text).The inclusion of this
Cairo of Ranucci’s is somewhat open to question, given that the diagram
consists of two pentagons, rather than the given ‘standard model’ of one.
Nonetheless, it is of interest due to the first example of this type.
18.
1971 Dunn, J. A.
‘Tessellations with Pentagons’. The Mathematical Gazette, Vol. 55, No.
394 (December, 366-369)
Finally, if the sides are all equal and x = x’ =90°, the tessellation in
Figure 5 becomes Figure 6 which is shown in Cundy and Rollett and is a
favourite street-tiling in Cairo.
The geometry of this basic pentagon is shown in Figure 7.
Of the utmost significance; the first recorded attribution.
19. 1972 Williams, Robert. The
Geometrical Foundation of Natural Structure. A Source Book of Design. Dover
Publications, Inc. 1979. Another edition, of another name, was of 1972.
Cairo
diagram (but not attributed) p. 38 in the context of the Laves tilings. This is
also interesting in that it shows ‘minimum and maximum’ values of the tiling,
of a square, and two rectangles (basketweave). The source of the D.G. Wood reference.
20. 1974
Clemens, Stanley
R. ‘Tessellations of Pentagons’. Mathematics Teaching, No. 67 (June), pp.18-19,
1974
Cairo
diagram (but not attributed) p. 18. Interesting in that this credits MacMahon
as the discover of the equilateral pentagon (p. 19), although this is not
substantiated. Likely, reading from MacMahon’s book, he just assumed this.
21. 1975
Parker, John. ‘Tessellations’, Topics, Mathematics Teaching 70, 1975, p.
34
Building on Clemens, immediately above, as noted by Parker. Loosely a Cairo
diagram (but not attributed) p. 34.
22.
1975 Gardner,
M. Scientific American. Mathematical Games, July. ‘On tessellating the
plane with convex polygon tiles’, pp. 112-117 (pp. 114, 116 re Cairo pentagon)
Gardner Quote Scientific American 1975 ‘On Tessellating the Plane with Convex
Tiles’, pp. 112-117
P. 114:
The most remarkable of all the pentagonal patterns is a tessellation of equilateral pentagons [‘c’]. It belongs only to Type 1*. Observe
how quadruplets of these pentagons can be grouped into oblong hexagons, each
set tessellating the plane at right angles to the other. This beautiful
tessellation [of equilateral
pentagons] is frequently seen as a street tiling in Cairo,
and occasionally on in the mosaics of Moorish buildings.
*errata (September 1975?) corrects this to Types 2 and 4
Gardner
then gives the construction:
The equilateral pentagon is readily constructed with a compass and
straightedge….
(What I refer to as the ‘45° construction’)
The second recorded attribution, based upon Dunn's account.
23. 1976 Odier,
Marc G. ‘Puzzle with Irregular Pentagonal Pieces’. United States Patent
3,981,505 21 September 1976
Cairo tile diagram Fig. 3,
and various patches of tiles formed with the pentagons.
24.
1976 O’Daffer, Phares. G; Clemens, Stanley
R. Geometry. An Investigative Approach 1^{st} edition, 2^{nd}
edition 1992 Addison-Wesley Publishing Company. (Note that I have the 2^{nd }edition,
not the 1^{st})
While a regular pentagon will not tessellate the plane, it is interesting to
note that there is a pentagon (see region A in Fig. 4.15) with all sides
congruent [i.e. equilateral] (but with different size angles) that will
tessellate the plane. A portion of this tessellation is shown in Fig. 4. 15. If
four of these pentagonal regions are considered together (see Region B), an
interesting hexagonal shape results that will tessellate the plane.
Cairo
diagram (but not attributed) p. 95 (text continues to p. 96).
25.
1977 Mottershead,
Lorraine.
Sources of Mathematical Discovery. Basil Blackwell.
Cairo
diagram (but not attributed) pp.106-107 on a chapter on tessellations, and a
subset of irregular pentagons.
Of note is the use of the Cairo tiles as a letter puzzle; although this is not
original with Mottershead, as perhaps might appear at first sight (as I did
myself to 2012). Although titled ‘… by Croton’, no further detail of ‘Croton’
is given. This diagram has now been determined as to appearing in The Listener, as detailed above, see 5.
1951 and 6. 1954. Unfortunately, the determination as to which types of
pentagon are here is fraught with difficulty due to such a small scale drawings
and the accuracy of the drawing is also in question, of which I am not prepared
to be categorical as to the type of pentagon here. They could be equilateral,
or near.
26.
1977 Schattschneider, Doris; Walker, W. M. C. Escher
Kaleidocycles. Tarquin Publications. First edition, 1977; I have the ‘special
edition’ of 1982.
One of Escher’s favourite geometric patterns was the tiling by pentagons
shown (Figure 35).These pentagons are not regular since their angles are not
all equal.
Cairo diagram (but not attributed) p. 26, also see p. 34, in the context of
a dodecahedron tiling decoration and Escher’s ‘Flower’, PD 132.
The type of Cairo
tiling is not explicitly stated; certainly it is of a 4, 1 type, likely of the dual
of the 3. 3. 4. 3. 4 type (90°, 120°), but
Escher did not use this!
27.
1978 Schattschneider, D. Tiling the Plane with Congruent Pentagons’ Mathematics Magazine. ‘Vol.1, 51, No.1 January 1978. 29-44.
P. 3
Three of the oldest known pentagonal tilings are shown in FIGURE 1. As
Martin Gardner observed in [5], they possess ‘unusual symmetry’. This symmetry
is no accident, for these three tilings are the duals of the only three
Archimedean whose vertices are valence 5. The underlying
Archimedean tilings are shown in dotted outline. Tiling (3) (dual of the 3.
3. 4. 3. 4) of FIGURE 1 has special
aesthetic appeal. It is said to appear as a street paving in Cairo [likely
referring to Martin Gardner or James Dunn’s quote; both authors are mentioned
in the bibliography]; it is the cover illustration for Coxeter’s Regular
Complex Polytopes [apparently equilateral], and was a favorite
pattern of the Dutch artist, M.C. Escher [square based intersections].
Escher’s sketchbooks reveal that this tiling is the unobtrusive geometric
network which underlies his beautiful; ‘shells and starfish’ pattern. He also
chose this pentagonal tiling as the bold network of a periodic design which
appears as a fragment in his 700 cm. Long print ‘Metamorphosis II’.
Tiling (3) can also be obtained in several other
ways. Perhaps most obviously it is a grid of pentagons which is formed when two
hexagonal tiles are superimposed at right angles to each other. F. Haag noted
that this tiling can also be obtained by joining points of tangency in a circle
packing of the plane [12]. It can also be obtained by dissecting a square into
four congruent quadrilaterals and then joining the dissected squares together
[26]. The importance of these observations is that by generalising these
techniques, other pentagonal tiles can be discovered.
The third recorded attribution, but not of a firsthand sighting. Of note
is Schattschneider’s care as to attribution, stating ‘it is said to appear as a
street tiling..’, likely as she had not seen a in situ picture, and so did not
sate so categorically that it was a
street tiling.
28.
1978 Pearce, Peter and Pearce, Susan. Polyhedra
Primer. Dale Seymour Publications
Cairo
diagram (but not attributed) on p. 35 and in the context of the dual tilings of
the semiregular tilings, p. 39. Decidedly lightweight, no discussion as such.
29.
1978 Lockwood, E.H; R.H. Macmillan. Geometric
symmetry. Cambridge University Press (and reprint 2008).
‘Indirect’ Cairo
reference p. 88
… are patterns [semi regular] of congruent pentagons such as are
often used for street paving in Moslem countries.
The inclusion of this book is somewhat of a moot point, in that Cairo
tiles are described very loosely here. However, as it is by Macmillan, this
rather fragmentary account is worthy of note, and curiously it does not
strictly tally with his later Mathematical
Gazette article.
30.
1979 Macmillan, R.H. Mathematical Gazette, 1979.
‘Pyramids and Pavements: some thoughts from Cairo’,
pp. 251-255
On a recent visit to Cairo
I was struck by two matters [concerning the pyramids
and pentagon tiling]…
and
P. 253
A pentagonal tessellation
Many of the streets of Cairo
are paved with a traditional Islamic tessellation of
pentagonal tiles, as shown in Fig. 4. The pentagons are all identical in size
and shape, having four sides equal and two of their angles 90°, as shown in
Fig. 5, where angles (* and *) and lengths (a and b) are marked. The tiles are
often in two colours, as in Fig. 4, and their pattern can then be classified as
belonging to the plane dichromatic symmetry group p4’ g’m. By making all those
tiles with a particular orientation of a single colour a polychromatic symmetry
pattern, of group p4^{(4)}, would be achieved; by an alternative
colouring it would be also be possible to produce a symmetry of group p4^{(4)}mg
^{(4)}, but I have never seen either of these actually used. (See [1],
p.89, Fig. 13.12.)
It
will be seen that the pattern formed by the tile edges can also be taken as two
interlinked and identical meshes. The question of interest is what may be the
possible variations in the shape of these pentagons and hexagons. We can see
that the slope of line CD in Fig. 4 can be varied, provided that the other
dimensions are altered suitably. The geometric conditions to be satisfied are
seen from Fig. 5 to be as follows:…..
P. 255
(iv) If * is such that , in Fig. 4, AB and CD are collinear, the
tessellation is particularly pleasing to the eye, and this is in fact the
proportion (108. ) often adopted in Cairo…
The fourth recorded attribution. Of note as to the depth of detail
Macmillan gives. Notably, he describes an in situ pentagon possessing of collinearity properties. A firsthand sighting, the second of only
two, and so of the utmost significance; also see Dunn. No reference is made in
the article itself or the references to any of the three above articles, and so
this is likely an independent account, as a ‘discovery’. As such, this is a
little surprising, in that Dunn’s article was also from the Mathematical
Gazette! Of note is the reference to the tiles being coloured, or arranged
of the same colour, ‘back to back’, this being the first recorded instance;
indeed, the only one!
31.
1982 Martin, George E. Transformation Geometry: An
Introduction to Symmetry, p. 119
The beautiful Cairo
tessellation with a convex equilateral pentagon as its prototile is illustrated
in Fig. 12.3. The tessellation is so named because such tiles were used for
many streets in Cairo.
Gives the ‘45°’ construction.
32.
1982 Murphy, Patrick. Modern Mathematics Made Simple.
Heinemann London Tessellations, Chapter 10, pp. 194-205, 262.
Cairo
diagram, of equilateral pentagons (but not attributed) p. 200.
33. 1983 Willson, John. Mosaic and Tessellated Patterns. How to
Create Them. Dover Publications, Inc. 1983. Plate 3Cairo tiling plate 3. (Neglected until 7 May 2013!)
34.
1984 Blackwell, William, Geometry in Architecture,
Wiley 1984, p. 54
...appearance of interlocking hexagons but consists of identical equal sided
(but not equal angular) pentagons. The hexagonal patterns cross at right angles
and the while pattern can be fit into a square or subdivided into modular
squares. This unusual pattern, which is seen in street tiling in Cairo
and occasionally in the mosaic of Moorish buildings.
The latter part of the quote is taken from Martin Gardner, word for
word.
35.
1986 McGregor, James and Watt, J. The Art of
Graphics for the IBM PC, pp 196-197
The plane cannot be tesselated (sic) by regular pentagons, but there
are an a number of irregular pentagons that will tessellate the plane. An
example of a pentagon that will tesselate (sic) is the well-known Cairo
tile, so called because many of the streets of Cairo
were paved in this pattern (Fig. 5.2): The Cairo
tile is equilateral but not regular because its angles are not all the same.
A minor part of a chapter on tessellations. Diagram p. 197.
36.
1986 Ehud, Bar-On. ‘A Programming Approach to
Mathematics’. Computers and Education. Elsevier. … then the possible
ways of tiling with pentagons are explored, especially the Cairo
tiling
Inconsequential reference. No diagrams are shown
37.
1986 Loeb. A. L. 'Symmetry and Modularity'. Computers
and Mathematics with Applications, Elsevier
38. 1986 Kappraff, Jay. ‘A Course in the Mathematics of
Design’. Computers and Mathematics with
Applications Vol. 12B, Nos. 3/4, pp 913-948
Cairo tiling in the context
of the set of 11 Laves tiling; p. 923 but
as such, inconsequential.
39.
1987 Grünbaum, Branko; Shephard, G. C. Tilings and
Patterns. W. H. Freeman and Company
For an account of a street tiling with pentagonal tiles common in Cairo
(Egypt) see Macmillan [1979]
P. 5, no discussion, just a reference to Macmillan’s article.
40.
1987 Burn, Bob. The Design of Tessellations. Cambridge
University
Press. Sheet 30. Equilateral pentagon.
41.
1989 Seymour, D; Britton, J. Introduction to
Tessellations. Dale Seymour Publications Cairo tiling (but not attributed)
p. 39.
The exact pentagon not described, almost certainly the
dual of the 3. 3. 4. 3. 4 (90°,
120° type).
Lightweight.
42.
1989 Harpe, P.
De La. ‘Quelques Problèmes Non
Résolus en Géométrie Plane’. L’Enseignement Mathématique, t 35 (1989), pp.
227-243 (in French)
Cairo tiling
on p. 232, likely taken from George Martin, given that it is the same ‘unusual’
configuration
43. 1989. Senechal, M. ‘Symmetry
Revisited’. Computers and Mathematics with
Applications. Vol 17, No. 1-3, pp1-12. 1989
Cairo
diagram in the context of the set of 11 Laves diagrams, p. 9; as such per se,
inconsequential.
44.
1989 Chorbachi, W. K. ‘In the Tower
of Babel:
Beyond Symmetry In Islamic Design’. Computers and Mathematics with
Applications. Vol. 17, No. 4-6, pp 751-789 (Cairo aspects 783-794), 1989
(reprinted in I. Hargittai, ed. Symmetry 2: Unifying Human
Understanding, Pergamon, New York, 1989.
The pattern of a favorite street tiling in Cairo
(US
spelling of favourite, note that Chorbachi also omits the dash between
favourite and street)
Fig. 19.16c _{2-3.}^{ }Two
different semiregular pentagons are drawn at the bottom of the page. On the
right side is the Islamic^{ }pentagon, where * is the critical value in
the design. On the left is the Western one given by J. A. Dunn in an article on
‘Tessellations with pentagons’ [30]. Dunn’s pentagon has an isosceles pentagon
triangle that has a critical length * for the two equal sides while the third
side is a or any given length. This tiling (Fig. 19.16c _{1}) is
referred to as the ‘favorite street tiling in Cairo’. In it, the tessellation
is considered hexagonal, each hexagon being a combination of four semi regular pentagons.
However, this tessellation is based on the 4-fold rotation of the semi regular
pentagon, with sides equal to two units and two opposite right angles. The
latter combination permits the 4-fold rotation of symmetry group 244 or p4g
Has interesting Cairo tiling references, pp. 783-784, and quotes James Dunn’s
1971 article, and beyond any reasonable doubt the quote given by Chorbachi is
taken from him as well. Equilateral pentagons.
Has references to ‘semi regular pentagons’ which is surely wrong terminology; I
had a web search for this, but I couldn't find references.
45. 1989 Hargittai, Istvan.
Symmetry 2, Unifying Human Understanding. Volume 2, Source of Chorbachi
article, see above pp. 783-794.
Not seen, Google Books reference.
46.
1990 Hill, Francis S. Jr. Computer Graphics.
Macmillan Publishing Company, New
York, P. 145.
An equilateral pentagon can tile the plane,
as shown in Figure 5.4. This is called a Cairo tiling because many streets in
Cairo were paved with tiles using this pattern. Note that this figure can also
be generated by drawing an arrangement of overlapping (irregular) hexagons.
Likely quoting from McGregor and Watt, given that the text is very much
alike, and their work is quoted and illustrations are used in the book.
47. 1991 Fetter, Ann E et al.
The Platonic Solids Activity Book. Key Curriculum Press/Visual Geometry
Project. Backline Masters.
Regular pentagons don’t tile, but many equilateral (though not equiangular)
pentagons do. [A Cairo tiling diagram is then shown.] This pattern is
seen in street tiling in Cairo and in the mosaics of Moorish buildings. A
similar tiling can be obtained of the dual of a semi regular tiling (see
exercise 8)
Cairo tiling pp. 21 and 97 (the latter of which repeats, as student
activities)
Almost certainly this quote is taken from Gardner, as detailed above.
48.
1991 Wells, David. The Penguin Dictionary of Curious
and Interesting Geometry. Penguin Books
P. 23: So called because it often appears in the streets of Cairo,
and in Islamic decoration. It can be seen in many ways, for example as cross
pieces rotated about the vertices of a square grid, their free ends joined by
short segments, or as two identical tessellations of elongated hexagons,
overlapping at right angles. Its dual tessellation, formed by joining the
centre of each tile to the centre of every adjacent tile, is a semiregular
tessellation of square and equilateral triangles.
P. 61: …Thus the dual of the tessellation of squares and equilateral
triangles is the Cairo
tessellation.
P. 177: The regular pentagon will not tessellate. Less regular
pentagons may, as in the Cairo
tessellation….
The first line of p. 23 bears resemblance to Gardner's
quote.
49. 1991 Kappraff, Jay. Connections The Geometric Bridge Between Art and Science. McGraw-Hill. p. 181
Shown as the dual of 3. 3. 4. 3. 4 tiling. Poorly executed diagram, with four different pentagons! However, the intention, due to an accompanying diagram, is indeed clear.
50. 1994 Leathard, Audrey. Going
inter-professional: working together for health and welfare
In the Cairo
tessellation (Wells 1991), dual tessellations are formed by overlaying a second
grid rotated 90 degrees to the first…P. 45:
Not seen, Google Books reference. Note that
this reference is only included for the sake ‘of everything’; the book is apparently
of a non-mathematical nature, and is not illustrated with the tiling. Quotes
the Wells reference.
51.
1994 Bays, Carter. Complex Systems Publications, Volume 8, Issue
2, 127-150, Cairo
aspect p. 148
‘Cellular
Automata in the Triangular Tessellation’
… the Cairo tessellation (a tiling of identical equilateral
pentagons)…
Cursory mention in passing.
52.
1997 Serra, Michael. Discovering Geometry: An
Inductive Approach. Key Curriculum Press, p. 404
The Cairo street
tiling shown at right is a very beautiful tessellation that uses equilateral pentagons (the sides are congruent but not the
angles). The pentagon is shown below right, with angle measures that will help
you draw your ...
Not seen, Google Books reference.
53. 1998 Singer, David A. Geometry
Plane and Fancy, 1998, p.34. Springer-Verlag
One particularly elegant tiling of the plane by pentagons is known as the
Cairo tessellation, because it can be seen as a street tiling in Cairo. The
pentagon used for this tiling can be constructed using straight edge and
compass… although it is not regular, it is equilateral…
Not seen, Google Books reference.
54. 1999
Stewart, Ian. ‘The Art of Elegant Tiling’. Scientific American. July
1999, pp. 96-98
Minor instance of coloured Cairo
tiling, p. 97, as devised by Rosemary Grazebrook
55. 1999 Beyer, Jinny. Designing
Tessellations, Contemporary Books, p. 144.
Lightweight.
56. 2001
Duffy, Edward, Greg Murty,
Lorraine
Mottershead. Connections Maths 7. Pascal
Press, p. 83
Cairo
streets have this Islamic pattern
Not seen, Google Books reference.
57.
2003 Teacher’s Guide: Tessellations and Tile
Patterns, p. 30 (Cabri) Geometric investigations on the VoyageTM 200 with
Cabri. Texas Instruments Incorporated
….Probably the most famous of these pentagonal patterns is the ‘Cairo
Tessellation’ named after the Islamic decorations found on the streets of
Cairo…Begins by quoting David Wells’ book … Curious… and likely the
text is based on his reference. However, the ‘Teacher’s Guide’ gives a
different tiling, interestingly a ‘collinear’
pentagon.
58.
2003 Gorini, Catherine A. The Facts on File Geometry
Handbook. 2003, 2009 revised edition. Facts on File Inc, and imprint of
Infobase publishing
Cairo tiling illustrated p. 22, equilateral. Gives the following definition:
Cairo tessellation: A
tessellation of the plane by congruent convex equilateral pentagons that have
two nonadjacent right angles; so called because it can be found on streets in
Cairo.
Oddly, Gorini shows an accompanying picture of a pentagon that is not
equilateral, a 4, 1 type…
59.
2003 Pritchard, Chris. The Changing Shape of
Geometry: Celebrating a Century of Geometry and Geometry Teaching
Is a favourite street tiling in Cairo
pp. 421-427. This is an anthology, and simply repeats Dunn’s article,
and follow-up correspondence. Nothing original is shown.
Not seen, Google Books reference.
60. 2003 Weisstein, Eric W. CRC
concise encyclopedia of mathematics, p. 313
A tessellation appearing in the streets of Cairo
and in many Islamic decorations. Its tiles are obtained by projection of a
dodecahedron, and it is the dual tessellation of the semiregular tessellation
of squares and equilateral triangles.
Interesting in that Weisstein defines this as the ‘projection of a
dodecahedron’ before the dual.
61. 2004
Parviainan, Robert. ‘Connectivity Properties of Archimedean and Laves
Lattices’. Uppsala
Dissertations in Mathematics 34. p. 9. 2004.
The lattice D (3^{2}.
4. 3. 4) is sometimes called the Cairo lattice, as the pattern occurs frequently as
tilings on the streets of Cairo.
A fleeting mention in the context of a study on
Laves tilings.
62.
2005 Mitchell, David. Sticky Note Origami: 25
Designs to make at your desk, Sterling Publication Company
The Cairo
Tessellation is an attractive and intriguing pattern of
tiles named as a result of its frequent occurrence on the streets of Cairo
and in other Islamic centers and sites. Cairo tiles are a special kind of
pentagon that unlike ordinary regular pentagons will fit together without
leaving gaps between them. Four of these slightly squashed pentagonal tiles
will from a stretched hexagon in the final pattern, stretched hexagons laid in
a vertical direction intersect other stretched hexagons laid horizontally
across and through them. If you make the tiles in four different colours the resulting
pattern is particularly interesting and attractive.
Mitchell doesn’t state exactly what type of Cairo tiling he is referring
to. However, upon checking his diagram, p. 58 it would appear to be equilateral.
However, due to the small scale nature, this is not categorically so.
Not seen, Google Books reference.
63.
2005 Phillips, George McArtney. Mathematics Is Not a
Spectator Sport. P. 193. Springer
Problem 6. 5. 3 Construct a dual of the 3. 3. 4. 3. 4 tessellation by
joining the centers of adjacent polygons. This is called the Cairo tessellation.
Observe that it has a pentagonal motif that has four sides of one length and
one shorter side
Not seen, Google Books reference.
64.
2005 Wilder, Johnston-Sue; John Mason. Developing
Thinking in Geometry, P. 182.
… is often referred to as the Cairo
tessellation as it appears in a mosque there.
Although the diagram is too small
in scale to measure with certainty, it appears to be of the dual of the 3. 3. 4. 3. 4 (90°,
120° type).
Not seen, Google Books reference.
65. 2005
Bays, Carter. Complex Systems
Publications, Volume 15, Issue 3, 245-252, Cairo aspect p. 249-250
‘A Note on the Game of Life in Hexagonal and Pentagonal Tessellations’
‘Here we have chosen the Cairo tiling…’
A curiosity, with the Cairo tiling acting as backdrop on the Game of Life.
66. 2005 Garcia, Paul. ‘The
Mathematical Pastimes of Major Percy Alexander MacMahon. Part 2 triangles and
beyond’. Mathematics in Schools,
September 2005, 20-22. PDF
Contains a Cairo tiling of sorts, p. 22
67. 2006 Sharp, John. ‘Beyond Su Doku’.
Mathematics Teaching in the Middle Years.
Vol. 12, No. 3 October 2006 pp. 165-169
Cairo
tiling on pp. 167-169, in the context of a ‘Cairo Su Doku’.
68.
2007 B. G. Thomas and M. A. Hann. in Sarhangi, Reza
(Ed). Bridges. Mathematical Connections in Art, Music, and Science.
(Ninth) Conference Proceedings 2007. Donostia, Spain. Patterned Polyhedra:
Tiling the Platonic Solids
…without gap or overlap. There are however
various equilateral pentagons that can tessellate
the plane. Probably the best known is the Cairo tessellation,
formed…
** 2007 Ollerton, Mike. 100+ Ideas for Teaching Mathematics p.66
This tessellation not only begs interesting questions about angle sizes and side ... The Cairo tessellation... A The challenge is to use this tile to fill 2D space.
69.
2008 Anon. Key Curriculum Press. Chapter 7
Transformations and Tessellations, p. 396
The beautiful Cairo street
tiling shown below uses equilateral pentagons.
This also gives a construction, of the well known ‘45° type’.
70.
2008 Goos, Merrilyn et al. Teaching Secondary School
Mathematics: Research and Practice for the 21^{st} Century.
The particular tiling pattern of an irregular pentagon, shown in Figure
9.16, is called the Cairo tessellation because it appears in a famous mosque in
Cairo.
Not seen, Google Books reference.
71.
2008 Thomas, B. G. and M. A. Hann. In Sarhangi, Reza
(Ed). Bridges. Mathematical Connections in Art, Music, and Science.
(Tenth) Conference Proceedings 2008. Leeuwarden,
Netherlands
There are, however, equilateral convex
pentagons that do tessellate the plane, such as the well known Cairo
tessellation shown in Figure 1.
Also, other minor references essentially in passing.
Cairo
reference and diagram p. 102 in ‘Patterning by Projection: Tiling the
Dodecahedron and other Solids’ gives an equilateral pentagon.
72.
2008 Fathauer, Robert. Designing and Drawing
Tessellations, p. 2.
A common street paving in Cairo,
Egypt
is shown above left. It is notable for the interesting tessellation formed by
pentagons, four of which form larger hexagons, with hexagon patterns running in
two different directions
Type of pentagon: Equilateral. Has a brief discussion on tessellations
in the 'real world', p. 2, with many photos of brickwork and paving stone
tessellations, all except for the ‘Cairo Pentagon’ tiling, where although this
is discussed, he shows a line drawing. Presumably, the reason for this is that
he was unable to locate a photo.
73.
2008 Thomas, B. G. and M. A. Hann. In Bridges.
Mathematical Connections in Art, Music, and Science.
There are, however, equilateral convex pentagons that do tessellate the
plane, such as the well-known Cairo
tessellation shown in Fig. 1.
Type of pentagon: Equilateral (p. 101).
**. 2008 Kaltenmorgen, Birgit. Der mathematische Patchworker. (in German) Wagner, Gelnhausen; 1^{st} edition pp. 82-83
Fünfeck beim Cairo-Tiling
74.
2009 Kaplan, Craig. S. Introductory Tiling Theory
for Computer Graphics. Morgan & Claypool Publishers, p. 33
The Laves tiling [3^{2}. 4. 3. 4] is sometimes known as the ‘Cairo
tiling’ because it is widely used there. p. 103
Not seen, Google Books reference.
75. 2009. Ollerton, Mike. The mathematics teacher's handbook, p. 148
… use four different colours
to make the 'Cairo' tiling design.
Not seen, Google Books reference.
76.
2010 Alsina, Claudi and Roger B. Nelsen. Charming
Proofs: A Journey Into Elegant Mathematics. Dolciani Mathematical
Expositions
Another pentagonal tiling can be created by overlaying two non-regular
hexagonal tilings illustrated in Figure 10.6. This rather attractive monohedral
pentagonal tiling is sometimes called the Cairo
tiling, for its reported use as a street paving design in that city.
Cairo
diagram p. 163. The type of pentagon is not detailed; unfortunately, the
diagram is too small a scale to measure with certainty.
Not seen, Google Books reference.
77. 2011 Elwes, Richard.
Maths 1001: Absolutely Everything You Need to Know about Mathematics in 1001
Bite-Sized Explanations. Quercus, p. 109
…it adorns the pavements of that city’ (Cairo).
Although it would appear likely that a single pentagon is intended, this shows
two different, but roughly alike pentagons, of which I assume that it just a
careless drawing. Given that the type of pentagon Elwes is referring to is
unclear; no assessment as to type is made.
78. 2011 Abdul Karim
Bangura. African
Mathematics: From Bones to Computers University Press of America, 2011
A basketweave
tessellation is topologically equivalent to the Cairo pentagonal tiling…
Not seen, Google Books reference. Cursory mention
in passing.
79. 2011 Goldemberg, Eric. Pulsation in Architecture p.338
Housing Exhibition in Vienna, Austria Project Description The Cairo Pods gave SPAN ... The Cairo Tessellation, known in mathematics also as an example of ...
80. 2012 Gengnagel, Christoph (ed), A. Kilian, Norbert Palz, Fabian Scheurer
Computational Design
Modeling: Proceedings of the Design Modeling Symposium. Springer, p. 229...
…on the mathematical configuration of a Cairo tessellation
81 2013 Iwamoto, Lisa. Digital Fabrications: Architectural and Material Techniques
… project uses a pentagonal Cairo tessellation pattern, flexibly aggregated to yield multiple overall arrangements. Each vertical layer of the cell was ...
82. 2014 Gorski, Hans-Joachim
and Susanne Müller-Philipp. Leitfaden
Geometrie: Für Studierende der Lehrämter, Springer p. 186
N.B. An apparent Cairo
tiling in a 1923 paper by F. Haag, "Die regelmässigen Planteilungen und
Punktsysteme." Zeitschrift fur Kristallographie 58 (1923): 478-488,
Figure 13, in that it is frequently quoted as a pentagonal tiling is
misleading; it's not a pentagon, but rather a quadrilateral.
Web
References
For the sake of accuracy, I restrict the listings here to a few prime
mathematics sites:
Wolfram
MathWorld
A tessellation appearing in the streets of Cairo
and in many Islamic decorations. Its tiles are obtained by projection of a
dodecahedron, and it is the dual tessellation of the semiregular tessellation
of squares and equilateral triangles.
Wikipedia
In geometry, a pentagon tiling is a tiling of the
plane by pentagons. A regular pentagonal tiling on ... Tiling Dual Semiregular
V3-3-4-3-4 Cairo
Pentagonal.svg ...
Created
on: 9 September 2011. Revised and enlarged: 22 September, 26 October, 1
November 2011; 21 April, 12 May, 3 December (D. G. Wood), 10 December (R.
Parviainan) 2012. Wholesale revision 14 December 2012, with quotes added to Section
1, and colour coding removed from Sections 1 and 2, which was a little
contrived, and did not make for easy reading. Also a general tidy-up. 4 October 2013. 10 July 2014: Moore and Odier references added. 7 January 2015: Haag (2) and Laves references added, inexplicably omitted previously)