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Defining a Cairo-like Tiling

Although the Cairo tiling is well known and assumed by common consent, of a ‘standard model’ of a pentagon with adjacent 90° angles (invariable), of four equal sides, with a long or short base (variable) in relation to the sides, there are various problems with the variable and invariable aspects in defining a generic, catch-all Cairo tiling. Various authorities give a variety of ‘Cairo tilings’ that are not strictly of the in situ paving! Rather, the in situ model can be described as a single specific instance of a family of pentagon tilings that can be varied somewhat, in that the side lengths and angles are different, and yet still remain ‘Cairo-like’ in nature, although at its extremes this obviously departs seriously from the ‘standard model’. As far as I am aware, no authority has attempted such a catch-all instance, of which I now address. An obvious starting point is the in situ paving, shown below as a line drawing (Fig. 1). As an aside, this has properties of collinearity, not shown here.

Figure 1. A line drawing of the in situ paving

By its very nature, this naturally has all the attributes. However, as alluded to above, aside from the 90° angles, the other angles and side lengths are variable. In itself, this would not not normally cause a problem in that this can be simply described. For instance, Doris Schattschneider, a pentagon tiling authority, gave me this:

...the pentagon in this tiling is type 4 in the list of convex pentagons that tile the plane, characterized by the conditions that it has two non-adjacent angles that are 90 degrees, each enclosed by two equal sides. A stronger condition would be that all four of these sides are equal.

As such, all well and good. This does indeed cover the ‘standard model’. However, it is possible to have a Cairo tiling in which all the sides are of the same length, i.e. an equilateral pentagon (Fig. 2), and of which this definition excludes. Therefore, the definition should include what can be described as this ‘special instance’. Building on the core description by Schattschneider, to accommodate the equialteral possibility I add:

???

But how best to do this?

Figure 2. Equilateral pentagon tiling

Doris’s definition seems to exclude the equilateral possibility (hence my addition). It's easy to tie oneself in knots accommodating this when it is ‘directly’ included. Perhaps it is better to retain the ‘...four of these sides are equal’ discussion, with reference to the equilateral sided pentagon as a special case, as an exception, after the main definition? How about two definitions, one suitable for a mathematical academic journal (for interest’s sake), and one more popular, suitable for Rawi, possibly mentioning mirror symmetry, which is only implied by Doris’s conditions; it may be obvious to the mathematician but not to the general reader!


Although not directly related to definition matters, as above I have alluded to a family of pentagon tilings, below I show a table of such possibilities, in half degree increments,  in which the pentagon in, various from maxima to minima, degenerating to squares and rectangle (the latter in a basket weave configuration), as first outlined by Robert Macmillan, in 1979. Some of these are of more interest than others, with special properties. For instance, No. 61 shows the dual of the 3.3.4.3.4 tiling (Fig. 3), whilst No. 46 shows the Cordovan pentagon (Fig. 4).


Figure 3. Dual of the 3.3.4.3.4 tiling


Figure 4. Cordovan Pentagon


The Cordovan pentagon has the interesting property related to collinearity in a way, when the parhexagon side is extended, it can be seen to alight on a vertex (a feature not typically seen). There is also another instance that also has an alighting on a vertex, No. 31, of which the pentagon does not seem to have been noticed before or have a title (Fig. 5)!  May I be permitted to call it the Bailey pentagon?

Figure 5. Bailey pentagon


Of note concerning this 'simple' angle listing is that it does not include the in situ paving, which has more 'complex' angles, of 108° 43’ and 143° 13’. This also can be described as a 'special case', as it has the property of collinearity.




Interior Angles

Comments

1

‘90, 90, 180, 90, 90’

Rectangle

2

90.5, 90, 179, 90, 90.5


3

91, 90, 178, 90, 91


4

91.5, 90, 177, 90, 91.5


5

92, 90, 176, 90, 92


6

92.5, 90, 175, 90, 92.5


7

93, 90, 174, 90, 93


8

93.5, 90, 173, 90, 93.5


9

94, 90, 172, 90, 94


10

94.5, 90, 171, 90, 94.5


11

95, 90, 170, 90, 95


12

95.5, 90, 169, 90, 95.5


13

96, 90, 168, 90, 96


14

96.5, 90, 167, 90, 96.5


15

97, 90, 166, 90, 97


16

97.5, 90, 165, 90, 97.5


17

98, 90, 164, 90, 98


18

98.5, 90, 163, 90, 98.5


19

99, 90, 162, 90, 99


20

99.5, 90, 161, 90, 99.5


21

100, 90, 160, 90, 100


22

100.5, 90, 159, 90, 100.5


23

101, 90, 158, 90, 101


24

101.5, 90, 157, 90, 101.5


25

102, 90, 156, 90, 102


26

102.5, 90, 155, 90, 102.5


27

103, 90, 154, 90, 103


28

103.5, 90, 153, 90, 103.5


29

104, 90, 152, 90, 104


30

104.5, 90, 151, 90, 104.5


31

105, 90, 150, 90, 105

Bailey Pentagon

32

105.5, 90, 149, 90, 105.5


33

106, 90, 148, 90, 106


34

106.5, 90, 147, 90, 106.5


35

107, 90, 146, 90, 107


36

107.5, 90, 145, 90, 107.5


37

108, 90, 144, 90, 108

Elements of a regular pentagon

38

108.5, 90, 143, 90, 108.5


39

109, 90, 142, 90, 109


40

109.5, 90, 141, 90, 109.5


41

110, 90, 140, 90, 110


42

110.5, 90, 139, 90, 110.5


43

111, 90, 138, 90, 111


44

111.5, 90, 137, 90, 111.5


45

112, 90, 136, 90, 112


46

112.5, 90, 135, 90, 112.5

Cordovan Pentagon

47

113, 90, 134, 90, 113


48

113.5, 90, 133, 90, 113.5


49

114, 90, 132, 90, 114


50

114.5, 90, 131, 90, 114.5


51

115, 90, 130, 90, 115


52

115.5, 90, 129, 90, 115.5


53

116, 90, 128, 90, 116


54

116.5, 90, 127, 90, 116.5


55

117, 90, 126, 90, 117


56

117.5, 90, 125, 90, 117.5


57

118, 90, 124, 90, 118


58

118.5, 90, 123, 90, 118.5


59

119, 90, 122, 90, 119


60

119.5, 90, 121, 90, 119.5


61

120, 90, 120, 90, 120

Dual of 3.3.4.3.4

62

120.5, 90, 119, 90, 120.5


63

121, 90, 118, 90, 121


64

121.5, 90, 117, 90, 121.5


65

122, 90, 116, 90, 122


66

122.5, 90, 115, 90, 122.5


67

123, 90, 114, 90, 123


68

123.5, 90, 113, 90, 123.5


69

124, 90, 112, 90, 124


70

124.5, 90, 111, 90, 124.5


71

125, 90, 110, 90, 125


72

125.5, 90, 109, 90, 125.5


73

126, 90, 108, 90, 126


74

126.5, 90, 107, 90, 126.5


75

127, 90, 106, 90, 127


76

127.5, 90, 105, 90, 127.5


77

128, 90, 104, 90, 128


78

128.5, 90, 103, 90, 128.5


79

129, 90, 102, 90, 129


80

129.5, 90, 101, 90, 129.5


81

130, 90, 100, 90, 130


82

130.5, 90, 99, 90, 130.5


83

131, 90, 98, 90, 131


84

131.5, 90, 97, 90, 131.5


85

132, 90, 196, 90, 132


86

132.5, 90, 95, 90, 132.5


87

133, 90, 194, 90, 133


88

133.5, 90, 93, 90, 133.5


89

134, 90, 92, 90, 134


90

134.5, 90, 91, 90, 134.5


91

‘135, 90, 90, 90, 135’

Square


Page created 26 June 2020
I might just ad that although I have previously composed a page with a similar title ‘Defining a Cairo-Type Tiling’, from 2013, this strayed from the core purpose, including a much broader definition, such as more than one pentagon, and even further loosening, that in retrospect although well intended, has serious concerns. However, upon revisiting the definition aspect recently, in connection with a forthcoming article for Rawi journal, of which a definition is to be included, and upon related correspondence with George Baloglou, one of my many collaborators in Cairo tilings matters, I once more revisited the problem. Although it may be thought to be simple, there are various intricases that militate against a simple definition. As ever, an open invitation to the reader.

Does anyone care to give a defintion?
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