Cairo Tiling‎ > ‎

Minima to Maxima


UNDER CONSTRUCTION

Of interest is that the in situ model can be described as just one of a family of ‘Cairo-like’ tilings, as the pentagon can vary, or degenerate, with the same broad conditions, from a square to a rectangle (the latter as a basketweave configuration), hence minima and maxima. This feature was first explicitly outlined by Robert Macmillan in his five ‘special cases’ [*], as well as implied (and illustrated)  by Herbert C. Moore in his 1909 patent ‘Tile’ [*]. However, Macmillan’s account was not of an ‘all-encompassing’ list as such (as I attempt to show below), but rather of establishing the principle, and using instances of ‘special interest’ along the way, namely of the in situ model (with properties of collinearity), an equilateral pentagon and dual of the 3.3.4.3.4 tiling. These can indeed be considered as special cases. However, are there more? The term 'special cases’ is open to interpretation. Arguably, although the above can be described as of  ‘core value’, there would appear to be others that could be included in this listing. A case in point is the ‘Cordovan pentagon’ with angles of 112.5°, 90°, 135°, 90°, 112.5°, based on the ‘Cordovan proportion’, as (seemingly) first described by the Spanish architect Raphael De la Hoz Arderius, and popularized by Encarción Reyes Iglesias. There are different ways of looking at this. One is that it is derived from a regular octagon. Another is that it is a ‘mid-range’ pentagon, balanced between the minima and maxima. An open question is to whether there are others (and of which I believe there are) worthy of inclusion. That said, Macmillan’s list is indeed of the main core values. To this end, I have examined, in two different procedures, instances from minima to maxima. I believe there are other ways, too. However, my process at least is a start. This is what I term as ‘incremental angles’ and ‘interstices’. To clarify, ‘incremental angles’, gives a sequence, in ½° increments, whilst ‘interstices’, gives a drawing on unit square graph paper. Both processes find different pentagons; there is no overlap. However, none of these admits the equilateral, of which this is composed by mathematical reasoning.  Further, the in situ model is only found by the interstices process. To this end, I present my computations in a table format, in four presentations:

1. By angle increments

2. By interstices

3. By equal sides

4. By a combination of all three.


With such a listing, one can examine for patterns, both local, for any one table, and globally, with data for all three. For instance, it is immediately obvious that the Cordovan pentagon is midway (Table 1), at No. 46, and can be considered as the most 'average'. Another observation is the dual of the 3.3.4.3.4 tiling is at No. 61 (Table 1). Therefore, an open question is to the possibility in that is there anything ‘special’ about No. 31, on the other side of the 'dividing' line, separated by the same amount of distance, that has so far escaped everyone's attention? I believe there is, possibly by coincidence, though, and of which I have taken the liberty of making a land grab and calling it the Bailey pentagon!  

To be continued, of text and Illustrations.


Table 1

Cairo Tilings by ½° Incremental Angles


Interior Angles

Base to Side Length*

Comments

1

‘90, 90, 180, 90, 90’

-

Degenerates to Rectangle

2

90.5, 90, 179, 90, 90.5

LB, SS


3

91, 90, 178, 90, 91

LB, SS


4

91.5, 90, 177, 90, 91.5

LB, SS


5

92, 90, 176, 90, 92

LB, SS


6

92.5, 90, 175, 90, 92.5

LB, SS


7

93, 90, 174, 90, 93

LB, SS


8

93.5, 90, 173, 90, 93.5

LB, SS


9

94, 90, 172, 90, 94

LB, SS


10

94.5, 90, 171, 90, 94.5

LB, SS


11

95, 90, 170, 90, 95

LB, SS


12

95.5, 90, 169, 90, 95.5

LB, SS


13

96, 90, 168, 90, 96

LB, SS


14

96.5, 90, 167, 90, 96.5

LB, SS


15

97, 90, 166, 90, 97

LB, SS


16

97.5, 90, 165, 90, 97.5

LB, SS


17

98, 90, 164, 90, 98

LB, SS


18

98.5, 90, 163, 90, 98.5

LB, SS


19

99, 90, 162, 90, 99

LB, SS


20

99.5, 90, 161, 90, 99.5

LB, SS


21

100, 90, 160, 90, 100

LB, SS


22

100.5, 90, 159, 90, 100.5

LB, SS


23

101, 90, 158, 90, 101

LB, SS


24

101.5, 90, 157, 90, 101.5

LB, SS


25

102, 90, 156, 90, 102

LB, SS


26

102.5, 90, 155, 90, 102.5

LB, SS


27

103, 90, 154, 90, 103

LB, SS


28

103.5, 90, 153, 90, 103.5

LB, SS


29

104, 90, 152, 90, 104

LB, SS


30

104.5, 90, 151, 90, 104.5

LB, SS


31

105, 90, 150, 90, 105

LB, SS

Bailey Pentagon

32

105.5, 90, 149, 90, 105.5

LB, SS


33

106, 90, 148, 90, 106

LB, SS


34

106.5, 90, 147, 90, 106.5

LB, SS


35

107, 90, 146, 90, 107

LB, SS


36

107.5, 90, 145, 90, 107.5

LB, SS


37

108, 90, 144, 90, 108

LB, SS

Elements of a regular pentagon

38

108.5, 90, 143, 90, 108.5

LB, SS


39

109, 90, 142, 90, 109

LB, SS


40

109.5, 90, 141, 90, 109.5

LB, SS


41

110, 90, 140, 90, 110

LB, SS


42

110.5, 90, 139, 90, 110.5

LB, SS


43

111, 90, 138, 90, 111

LB, SS


44

111.5, 90, 137, 90, 111.5

LB, SS


45

112, 90, 136, 90, 112

LB, SS


46

112.5, 90, 135, 90, 112.5

LB, SS

Cordovan Pentagon

47

113, 90, 134, 90, 113

LB, SS


48

113.5, 90, 133, 90, 113.5

LB, SS


49

114, 90, 132, 90, 114

LB, SS


50

114.5, 90, 131, 90, 114.5

SB, LS


51

115, 90, 130, 90, 115

SB, LS


52

115.5, 90, 129, 90, 115.5

SB, LS


53

116, 90, 128, 90, 116

SB, LS


54

116.5, 90, 127, 90, 116.5

SB, LS


55

117, 90, 126, 90, 117

SB, LS


56

117.5, 90, 125, 90, 117.5

SB, LS


57

118, 90, 124, 90, 118

SB, LS


58

118.5, 90, 123, 90, 118.5

SB, LS


59

119, 90, 122, 90, 119

SB, LS


60

119.5, 90, 121, 90, 119.5

SB, LS


61

120, 90, 120, 90, 120

SB, LS

Dual of 3.3.4.3.4

62

120.5, 90, 119, 90, 120.5

SB, LS


63

121, 90, 118, 90, 121

SB, LS


64

121.5, 90, 117, 90, 121.5

SB, LS


65

122, 90, 116, 90, 122

SB, LS


66

122.5, 90, 115, 90, 122.5

SB, LS


67

123, 90, 114, 90, 123

SB, LS


68

123.5, 90, 113, 90, 123.5

SB, LS


69

124, 90, 112, 90, 124

SB, LS


70

124.5, 90, 111, 90, 124.5

SB, LS


71

125, 90, 110, 90, 125

SB, LS


72

125.5, 90, 109, 90, 125.5

SB, LS


73

126, 90, 108, 90, 126

SB, LS


74

126.5, 90, 107, 90, 126.5

SB, LS


75

127, 90, 106, 90, 127

SB, LS


76

127.5, 90, 105, 90, 127.5

SB, LS


77

128, 90, 104, 90, 128

SB, LS


78

128.5, 90, 103, 90, 128.5

SB, LS


79

129, 90, 102, 90, 129

SB, LS


80

129.5, 90, 101, 90, 129.5

SB, LS


81

130, 90, 100, 90, 130

SB, LS


82

130.5, 90, 99, 90, 130.5

SB, LS


83

131, 90, 98, 90, 131

SB, LS


84

131.5, 90, 97, 90, 131.5

SB, LS


85

132, 90, 196, 90, 132

SB, LS


86

132.5, 90, 95, 90, 132.5

SB, LS


87

133, 90, 194, 90, 133

SB, LS


88

133.5, 90, 93, 90, 133.5

SB, LS


89

134, 90, 92, 90, 134

SB, LS


90

134.5, 90, 91, 90, 134.5

SB, LS


91

‘135, 90, 90, 90, 135’

-

Degenerates to Square



* LB, SS = Long Base, Short Sides; SB, LS = Short Base, Long Sides

Table 2

Cairo Tilings by Square Grid Interstices


Interior Angles

Base to Side Length*

Comments

Bailey No

1

129.81, 90, 100.39, 90, 129.81

SB, LS

Near Rectangle

20

2

129.29, 90, 101.42, 90, 129.29

SB, LS


16

3

128.66, 90, 102.68, 90, 128.66

SB, LS


14

4

127.87, 90, 104.25, 90, 127.87

SB, LS


11

5

126.87, 90, 106.26, 90, 126.87

SB, LS


9

6

125.54, 90, 108.92, 90, 125.54

SB, LS


6

7

123.69, 90, 112.62, 90, 123.69

SB, LS


5

8

120.96, 90, 118.07, 90, 120.96

SB, LS


3

9

119.05, 90, 121.89, 90, 119.05

SB, LS


12

10

116.57, 90, 126.87, 90, 116.57

SB, LS


2

11

114.44, 90, 131.11, 90, 114.44

SB, LS


17

12

113.2, 90, 133.6, 90, 113.2

LB, SS


7

13

111.8, 90, 136.4, 90, 111.8

LB, SS


15

14

108.43, 90, 143.13, 90, 108.43

LB, SS

In Situ Model

1

15

105.26,  90, 149.49, 90, 105.26

LB, SS


19

16

104.04,  90, 151.93, 90, 104.04

LB, SS


10

17

101.31,  90, 157.38, 90, 101.31

LB, SS


4

18

99.46,  90, 161.08, 90, 99.46

LB, SS


21

19

98.13,  90, 163.74, 90, 98.13

LB, SS


8

20

96.34,  90, 167.32 90, 96.34

LB, SS


13

21

95.19,  90, 169.61, 90, 95.19

LB, SS

Near Square

18


* LB, SS = Long Base, Short Sides; SB, LS = Short Base, Long Sides


Cairo Tiling by Unique Equal Sides


Interior Angles

Base to Side Length

Comments

1

114.18, 90, 131.24, 90, 114.18

Equal

Equilateral sides, only one instance




Table 4

Cairo Tilings by ½° Incremental Angles, Interstices and Equilateral


Interior Angles

Base to Side Length*

Comments, Bailey No.

1

‘90, 90, 180, 90, 90’

-

Degenerates to Rectangle

2

90.5, 90, 179, 90, 90.5

LB, SS


3

91, 90, 178, 90, 91

LB, SS


4

91.5, 90, 177, 90, 91.5

LB, SS


5

92, 90, 176, 90, 92

LB, SS


6

92.5, 90, 175, 90, 92.5

LB, SS


7

93, 90, 174, 90, 93

LB, SS


8

93.5, 90, 173, 90, 93.5

LB, SS


9

94, 90, 172, 90, 94

LB, SS


10

94.5, 90, 171, 90, 94.5

LB, SS


11

95, 90, 170, 90, 95

LB, SS


12

95.19,  90, 169.61, 90, 95.19

LB, SS

B18

13

95.5, 90, 169, 90, 95.5

LB, SS


14

96, 90, 168, 90, 96

LB, SS


15

96.34,  90, 167.32 90, 96.34

LB, SS

B13

16

96.5, 90, 167, 90, 96.5

LB, SS


17

97, 90, 166, 90, 97

LB, SS


18

97.5, 90, 165, 90, 97.5

LB, SS


19

98, 90, 164, 90, 98

LB, SS


20

98.13,  90, 163.74, 90, 98.13

LB, SS

B8

21

98.5, 90, 163, 90, 98.5

LB, SS


22

99, 90, 162, 90, 99

LB, SS


23

99.46,  90, 161.08, 90, 99.46

LB, SS

B21

24

99.5, 90, 161, 90, 99.5

LB, SS


25

100, 90, 160, 90, 100

LB, SS


26

100.5, 90, 159, 90, 100.5

LB, SS


27

101, 90, 158, 90, 101

LB, SS


28

101.31,  90, 157.38, 90, 101.31

LB, SS

B4

29

101.5, 90, 157, 90, 101.5

LB, SS


30

102, 90, 156, 90, 102

LB, SS


31

102.5, 90, 155, 90, 102.5

LB, SS


32

103, 90, 154, 90, 103

LB, SS


33

103.5, 90, 153, 90, 103.5

LB, SS


34

104, 90, 152, 90, 104

LB, SS


35

104.04,  90, 151.93, 90, 104.04

LB, SS

B10

36

104.5, 90, 151, 90, 104.5

LB, SS


37

105, 90, 150, 90, 105

LB, SS

Bailey Pentagon

38

105.26,  90, 149.49, 90, 105.26

LB, SS

B19

39

105.5, 90, 149, 90, 105.5

LB, SS


40

106, 90, 148, 90, 106

LB, SS


41

106.5, 90, 147, 90, 106.5

LB, SS


42

107, 90, 146, 90, 107

LB, SS


43

107.5, 90, 145, 90, 107.5

LB, SS


44

108, 90, 144, 90, 108

LB, SS

Elements of a regular pentagon

45

108.43, 90, 143.13, 90, 108.43

LB, SS

B1, In situ model

46

108.5, 90, 143, 90, 108.5

LB, SS


47

109, 90, 142, 90, 109

LB, SS


48

109.5, 90, 141, 90, 109.5

LB, SS


49

110, 90, 140, 90, 110

LB, SS


50

110.5, 90, 139, 90, 110.5

LB, SS


51

111, 90, 138, 90, 111

LB, SS


52

111.5, 90, 137, 90, 111.5

LB, SS


53

111.8, 90, 136.4, 90, 111.8

LB, SS

B15

54

112, 90, 136, 90, 112

LB, SS


55

112.5, 90, 135, 90, 112.5

LB, SS

Cordovan Pentagon

56

113, 90, 134, 90, 113

LB, SS


57

113.2, 90, 133.6, 90, 113.2

LB, SS

B7

58

113.5, 90, 133, 90, 113.5

LB, SS


59

114, 90, 132, 90, 114

LB, SS


60

114.18, 90, 131.24, 90, 114.18

Equilateral

Equilateral

61

114.44, 90, 131.11, 90, 114.44

SB, LS

B17

62

114.5, 90, 131, 90, 114.5

SB, LS


63

115, 90, 130, 90, 115

SB, LS


64

115.5, 90, 129, 90, 115.5

SB, LS


65

116, 90, 128, 90, 116

SB, LS


66

116.5, 90, 127, 90, 116.5

SB, LS


67

116.57, 90, 126.87, 90, 116.57

SB, LS

B2

68

117, 90, 126, 90, 117

SB, LS


69

117.5, 90, 125, 90, 117.5

SB, LS


70

118, 90, 124, 90, 118

SB, LS


71

118.5, 90, 123, 90, 118.5

SB, LS


72

119, 90, 122, 90, 119

SB, LS


73

119.05, 90, 121.89, 90, 119.05

SB, LS

B12

74

119.5, 90, 121, 90, 119.5

SB, LS


75

120, 90, 120, 90, 120

SB, LS

Dual of 3.3.4.3.4

76

120.5, 90, 119, 90, 120.5

SB, LS


77

120.96, 90, 118.07, 90, 120.96

SB, LS

B3

78

121, 90, 118, 90, 121

SB, LS


79

121.5, 90, 117, 90, 121.5

SB, LS


80

122, 90, 116, 90, 122

SB, LS


81

122.5, 90, 115, 90, 122.5

SB, LS


82

123, 90, 114, 90, 123

SB, LS


83

123.5, 90, 113, 90, 123.5

SB, LS


84

123.69, 90, 112.62, 90, 123.69

SB, LS

B5

85

124, 90, 112, 90, 124

SB, LS


86

124.5, 90, 111, 90, 124.5

SB, LS


87

125.54, 90, 108.92, 90, 125.54

SB, LS

B6

88

125, 90, 110, 90, 125

SB, LS


89

125.5, 90, 109, 90, 125.5

SB, LS


90

126, 90, 108, 90, 126

SB, LS


91

126.5, 90, 107, 90, 126.5

SB, LS


92

126.87, 90, 106.26, 90, 126.87

SB, LS

B9

93

127, 90, 106, 90, 127

SB, LS


94

127.5, 90, 105, 90, 127.5

SB, LS


95

128, 90, 104, 90, 128

SB, LS


96

128.5, 90, 103, 90, 128.5

SB, LS


97

129, 90, 102, 90, 129

SB, LS


98

129.29, 90, 101.42, 90, 129.29

SB, LS

B16

99

129.5, 90, 101, 90, 129.5

SB, LS


100

129.81, 90, 100.39, 90, 129.81

SB, LS

B20

101

128.66, 90, 102.68, 90, 128.66

SB, LS

B14

102

127.87, 90, 104.25, 90, 127.87

SB, LS

B11

103

130, 90, 100, 90, 130

SB, LS


104

130.5, 90, 99, 90, 130.5

SB, LS


105

131, 90, 98, 90, 131

SB, LS


106

131.5, 90, 97, 90, 131.5

SB, LS


107

132, 90, 196, 90, 132

SB, LS


108

132.5, 90, 95, 90, 132.5

SB, LS


109

133, 90, 194, 90, 133

SB, LS


110

133.5, 90, 93, 90, 133.5

SB, LS


111

134, 90, 92, 90, 134

SB, LS


112

134.5, 90, 91, 90, 134.5

SB, LS


113

‘135, 90, 90, 90, 135’

-

Degenerates to Square


* LB, SS = Long Base, Short Sides; SB, LS = Short Base, Long Sides

References

Dunn, James. ‘Tessellations with Pentagons’. The Mathematical Gazette, Vol. 55, No. 394 December 1971, pp. 366-369

Gardner, Martin. Scientific American. Mathematical Games, July. ‘On tessellating the plane with convex polygon tiles’, pp. 112-117 (p. 114 and 116 re Cairo aspect), 1975

Macmillan, Robert H. Mathematical Gazette, 1979. ‘Pyramids and Pavements: some thoughts from Cairo’, pp. 251-255.

MacMahon, Percy A. New Mathematical Pastimes. Cambridge University Press 1921 and 1930. (Reprinted by Tarquin Books, 2004)

Raphael De la Hoz Arderius. 'La proporción cordobesa'. Acta VII  Jornadas  Andaluzas  de Educación  Mathemática. Thales, 1996, pp. 67-84 (NOT SEEN)

Schattschneider, Doris. Mathematics Magazine, January. ‘Tiling the Plane with Congruent Pentagons’, pp. 29-44 (p. 30 re Cairo aspect).

Moore, Herbert C. United States Patents 928,320 and 928,321 of 20 July 1909



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