median
don steward
mathematics teaching 10 ~ 16

## Wednesday, 19 October 2011

### trisides

a long time ago, Geoff Giles (of DIME materials fame) made reference to transforming shapes that tessellate into triangles
he called these 'trisides'

the theory is that all tessellations are actually tessellations of triangles in the background

the sections of lines between two dots (corners of a triangle) have rotational symmetry, order 2

to create the 'trisides' from e.g a pentomino tessellation often requires going beyond the perimeter of a shape to place a dot (corner)

the sections of lines between two dots (the corners of a triangle) must have rotational symmetry, order 2

the whole of the perimeter needs to be encompassed within three sections, each section of which has rotational symmetry

here are some of the pentominoes, turned into 'trisides' :

each section between a pair of dots has rotational symmetry
the whole of the perimeter is encompassed by the three sections

the trisides so formed aren't unique

the resulting three dots form a triangle of area equal to the original shapes (5 squares)

four of the ways to turn this pentomino into a triside (there are other ways)

in trying to turn tessellating shapes into trisides, one technique is to start anywhere (within reason) and then adjust the outcome so that all three sections between the dots have rotational symmetry

another selection of pentominoes turned into trisides, as further examples:

the remaining four pentominoes also convert to 'trisides':

an implication is that by means of this 'triside' transformation, tessellations of pentominoes (which all tessellate) can be seen to have a triangle tessellation lying 'behind' them

furthermore, it seems that many if not all straight line bounded shapes that tessellate can be turned into 'trisides'

sometimes with a lot of perseverance....

tessellations of straight line bounded 'tiles' can be seen as a tessellation of triangles....

a chevron (hexagon) turned into a triside

so are all tessellations of straight line bounded shapes actually just tessellations of triangles?