Download 4.2 Tilings INSTRUCTOR NOTES

4.2 Tilings
INSTRUCTOR NOTES
1 hour 15 min.
Before class begins, write the table below on the board and have students who
arrive early fill in the angles (using the formula from section 4.1).
The table below gives the size of the interior angle in a regular polygon.
sides
3
4
5
6
7
8
9
10
11
12
angle
60˚
90˚
108˚
120˚
128.57˚
135˚
140˚
144˚
147.27˚
150˚
polygon
triangle
square
pentagon
hexagon
heptagon
octagon
nonagon
decagon
11-gon
dodecagon
Reference pictures (tilings) on pages 254, 261, 265.
p. 253 Edge-to-edge:
Each edge of a tile coincides exactly with one edge of a bordering tile.
Reference tilings
on page 253-4.
NOT Edge-to-edge
p. 253 Monohedral:
Tilings that use only one size and one shape of tile (up or down).
p. 254 Regular Tiling:
An edge-to-edge monohedral tiling where all the tiles are regular polygons.
1
Ask class, “What type of regular polygons can produce a regular tiling?”
Tell them…
NOTE: The sum of the interior angles surrounding
a vertex in any tiling must add up to 360°.
In a regular tiling all of the angle measures
must be the same. So the “number of
polygons” surrounding a vertex times the
measure of the same angle must equal 360°.
In general for a regular tiling,
(number of polygons)*(angle measure) = 360°
OR
(number of polygons) = 360°/(angle measure)
For example, in a tiling for regular hexagons we have each vertex
surrounded by 3 regular hexagons because
(number of polygons) = 360°/120° = 3.
However, note that the “number of polygons” must be an integer. Looking
at the table at the beginning and running through the possibilities we see that
the only possibilities for a regular tiling are those by regular triangles, squares
or regular hexagons.
Reference regular tilings at the bottom of page 254.
CLASS DO: p. 266 # 2 not a regular polygon (interior angles are not the same)
p. 257 Semiregular Tiling:
An edge-to-edge tiling by at least two regular
polygons with all vertices the same type.
2
Write on board…
REGULAR TILING
• Edge-to-edge
• ONE regular polygon of just one
size (monohedral)
• Only ones are
o Equilateral triangles
o squares
o regular hexagons
SEMIREGULAR TILING
• Edge-to-edge
• TWO OR MORE regular
polygons
• All vertices are the same type
p. 255 Vertex Type:
Two vertices are said to be of the same type if surrounded by
• same type of polygons
• same number of each type
• same order (clockwise or counterclockwise)
CLASS DO: p. 266 # 4 not edge-to-edge
all vertices are not the same type
6 all vertices are not the same type
YOU DO: p. 267 # 14a
HINT: Keep adding polygons while striving to make every vertex the same
type. Don’t go out in only one direction: try to wrap around.
Eventually you will arrive at a contradiction.
3
YOU DO: p. 267 # 16
HINT: Use table at the beginning and formula from section 4.1.
Answer: Regular nonagon and equilateral triangle.
“Shopping” for regular polygons using 360°.
CLASS DO: Use the table at the beginning to find other combinations for tilings
with regular polygons.
Possible answers are:
• square, 2 octagons
• 2 triangles, square, dodecagon
• square, hexagon, dodecagon
Also
• triangle, decagon, 15-gon
• triangle, octagon, 24-gon
4