4d + 4u

4u: using 4 lengths of length 1 unit
4d: and 4 lengths of length (square root 2) 1 diagonal
create a closed shape

try to create shapes that have reflection or rotational symmetry (or both)

powerpoint: 4d + 4u

see also du

bead patterns

continue and complete a pattern:

see frieze patterns

quartering a 5 by 5 grid

hopefully being systematic, using some logic/reasoning, students can look to find all the ways to
quarter a 5 by 5 dotty grid with
(i) rotational symmetry order 4
(ii) rotational symmetry order 2
(iii) one line of symmetry

they could also find all 13 ways to half a 4 by 4 grid

the powerpoint for this

order 4

order 4, rotational symmetry designs (11 by 11 grid):  

or more intricate ones like Jamal Muhsin's:
(mostly 25 by 25 grid)

(a copy, based on the one below)

or, for the even more adventurous, try some labyrinth tessellations

for order 4 patterns see Brian Eno's 77 million paintings (various locations)

Korean symbol symmetry

mostly from Chloe Koo
what symmetries do these symbols have?

quartering

task from 'the Little Book of Big Brain Games'
by Ivan Moscovich


the four shapes have to completely fill the space

some students might appreciate that each shape needs to have 2 squares (8 divided by 4) in the inner ring and 4 (16 divided by 4) in the outer ring - which makes this task a lot easier (thanks to Jenni Ingram, Oxford Uni. for this insight).

polyiamonds and rotational symmetry

a hexiamond is made with 6 equilateral triangles stuck together, edge to edge

find all 12 of the hexiamonds

or use these:

(these are given names in some sources)
















put two of the same hexiamond shapes together to create a shape with rotational symmetry

explore other options

or use these resources:

try to fit all 12 hexiamonds together to fill this shape

one solution

there are others....

55 altogether according to
David Goodger












thanks to recmath for these ideas and further information

symmetry patterns