don steward
mathematics teaching 10 ~ 16

Saturday, 11 January 2020

dissecting regular polygons into rhombuses

a ppt is here

find the angles in the rhombuses
(this involves knowing how to find the interior angle of a regular polygon)

only polygons with an even number of sides are considered (i.e. they are examples of 'zonogons')

the rhombuses will all have the same side length
pairs of opposite sides are parallel, so stripes of rhombuses are formed when you 'step' from one edge to one parallel to it - across a 'zone'

pattern blocks (or virtual ones) are good for exploring dodecagon dissections

three types of dissection methods:
  1. spreading out from a vertex ('lotus flower' or a 'fan')
  2. spanning out from the centre ('star or a 'rosette')
  3. adjacent edges paired, from the outside
type 1: growing from a vertex

the rhombus vertices are determined by the diagonals of the polygon

type 2, dissecting from the centre:

with diagonals drawn in
what happens for a sequence of angles?
type 3:
starting from pairs of edges on the regular (even-sided) polygon
sometimes this results in rhombuses at the centre and sometimes it does not

showing how this process does not work for a dodecagon

the 'lotus flower' arrangement can be viewed as a section of the 'star' arrangement

what happens to the angle sequences and why?

see Paul Stephenson's article in SYMetryplus 68 (spring 2019) here

No comments: