Plane symmetry under constraints

Introduction

Mathematicians know that there are 17 different combinations of transformations that produce symmetric patterns that cover the plane.  These are called the symmetries of the plane. Coxeter (1969, pp. 58-60) shows how six of these can be produced by tiling the plane with congruent rectangles. Many walls and pavements, however, make use of two sizes of rectangles, but does that allow other symmetries?  And what if the bricks are allowed to have any shape at all?  In that case any of the plane symmetries is possible, but are bricks manufactured that would produce all the possibilities?

Congruent rectangles Mixed rectangles Other shapes

Brick patterns made of Congruent Rectangles

The six brick patterns identified and classified by Coxeter are shown below. Two questions arise:
  • Are there any other symmetries of the plane that can be produced using only 1 by n congruent bricks?
  • Are there any other patterns that can be formed with 1 by n bricks other than those shown here?  If so which symmetry group do they belong to?

Symmetry cmm:

A half-turn and two perpendicular reflections:
  cmm

Running or Stretcher Bond
 stretcher ANS

Symmetry pmm:

reflections on the sides of a rectangle:

pmm

Stack Bond
 stack ANS

Symmetry p4g:

a reflection and a quarter-turn:

p4g

Basket Weave
 basket WNS

Symmetry p2:

three half turns:

p2

Raking Stretcher Bond
 WNS

Symmetry pmg:

a reflection and two half turns:

pmg
WNS

Symmetry pgg:

two perpendicular glide reflections:

pgg

Herringbone
 herringbone WNS

Brick patterns made of Mixed Rectangles

Brick patterns involving two sizes of rectangles are shown below, classified by symmetry group. 
  • Are there any other symmetries of the plane that can be produced using two sizes of rectangular bricks?
[cmm] [pmm] [p4] [p4g]


Symmetry cmm:

ANS

WNS

Symmetry pmm:

WNS

ANS

Symmetry p4:

Pinwheel p4CW
temp (Pinwheel)

Symmetry p4g:

Please contribute to this web site by sending me a photograph of this pattern.  Credit will be given.  <david.reid@acadiau.ca>


Brick patterns made of Non-Rectangular Bricks

Brick patterns using non-rectangular bricks can in theory belong to any of the 17 plane symmetry groups, but observing patterns made of bricks belonging to every group is not a trivial task.  You are invited to add to this category but sending photos to me at  <david.reid@acadiau.ca>
[pmg] [p4m]


Symmetry pmg:

  VBC

 

Symmetry p4m:

  VBC







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This page maintained by David A Reid, Email: david.reid@acadiau.ca


 
References

Coxeter, H. S. M. (1969). Introduction to geometry, 2nd Edition.  New York : Wiley.
 
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