median
don steward
mathematics teaching 10 ~ 16

## Tuesday, 22 December 2015

### 'Cairo' pentagon tilings

a tessellation of a single symmetric pentagon,
sometimes with (several) equal sides,
two (opposite) right angles
comes in various forms

found in Cairo as paving tiles
(not that old)

what relationship is there between the angles if two are right angles and two of the other (obtuse) angles are equal?

what if the remaining 3 of them are equal?

David Bailey (from Grimsby, England, with a keen interest in recreational mathematics) has undertaken a very thorough analysis of the variety and dates of 'in situ' tiles
so far the oldest dated version he has been able to establish is 1956

due to the two 90 degree angles in the pentagon
the tessellation(s) have a 'skeleton' (as David Wells calls them) of squares

so the variety of tessellations is created by the different angle in the rhombus:

the pentagon tessellation can be viewed with other square 'skeletons'

or another way
are these squares?

or with isosceles (at least) triangles

what relationship is there for the apex and two equal base angles?

or with trapeziums
what is the relationship here?

a version of the tiling can be created from the 4, 3, 3, 4, 3 semi-regular tessellation (as a dual)

what are the angles in the tiles?

are the tiles congruent?

a tessellation of Cairo-like tiles can be drawn on isometric paper

but...  there are two different types of pentagon here
establish that their angles are the same

David Bailey quotes Robert H Macmillan's claim that collinearity is a feature of some of the tessellations and explores this

four of the sides are the same length

what are the angles in each pentagon tile for this arrangement?

use trigonometry to establish the angles in each pentagon for this arrangement

what collinearity is there?

it is possible to have
all the sides the same length

what are the angles for this equilateral pentagon tile?

if the triangle formed by joining the apex of the pentagon to the two bottom corners is equilateral

there is a collinearity

the triangle shown is equilateral

find angle 'a'
and establish the (seeming) collinearity property