Title: Reptiles, Partridges, and Golden Bees: Tiling Shapes with Similar Copies
1Reptiles, Partridges, and Golden BeesTiling
Shapes with Similar Copies
- Erich Friedman
- Stetson University
- February 21, 2003
- efriedma_at_stetson.edu
2Perfect Tilings
3Tiling Rectangleswith Unequal Squares
- A rectangle can be tiled with unequal squares.
(Moron, 1925)
- There is a method of producing such tilings.
(Tutte, Smith, Stone, Brooks, 1938)
4Tiling Rectangleswith Unequal Squares
- Take a planar digraph where every edge points
down. - Find weights for the edges so
- the total distance from vertex to vertex is path
independent. - the flow into a vertex is equal to the flow out
of the vertex. - (these are just Kirchoffs Laws if each edge has
unit resistance.)
5Tiling Rectangleswith Unequal Squares
6Tiling Rectangleswith Unequal Squares
7Perfect Tilings
- A perfect tiling of a shape is a tiling of that
shape with finitely many similar but
non-congruent copies of the same shape.
- The order of a shape is the smallest number of
copies needed in a perfect tiling.
Are there perfect
tilings of squares?
8Perfect Square Tilings
- Mostly using trial and error, a perfect square
tiling with 69 squares was found. (Smith, Stone,
Brooks, 1938) - The first perfect tiling to be published
contained 55 squares. (Sprague, 1939) - For many years, the smallest possible order was
thought to be 24. (Bristol, 1950s)
9Perfect Square Tilings
- But eventually the smallest order of a perfect
square tiling was shown to be 21. (Duijvestijn,
1978)
10Perfect Square Tilings
The number of perfect squares of a given
order order number 21
1 22 8 23 12
24 26 25 160 26
441
- Open Problem How many perfect squares of order
27?
Are there perfect
tilings of all rectangles?
11Perfect Tilings of Rectangles
- There are perfect tilings of all rectangles since
we can stretch a perfect tiling of squares.
- The order of a 2x1 rectangle is 8 (Jepsen, 1996)
12Perfect Tilings of Rectangles
- Open Problem Is the order of a 3x1 rectangle
equal to 11? (Jepsen, 1996)
- Open Problem What are the orders of other
rectangles?
13New Perfect Tilings from Old
- If a shape S has a perfect tiling using n copies,
and a perfect tiling using m copies, it has a
perfect tiling using nm-1 copies. - Take an n-tiling of S, and replace the smallest
tile with an m-tiling of S.
14Perfect Tilings of Triangles
Do all triangles
have perfect tilings?
15Perfect Tilings of Triangles
- There are perfect tilings for most triangles,
into either 6 or 8 smaller triangles.
16Perfect Tilings of Triangles
- There is no perfect tiling of equilateral
triangles. - Consider the smallest triangle on the bottom.
- It must touch a smaller triangle.
- This triangle must touch an even smaller one.
- There are only finitely many triangles. QED
17Perfect Tilings of Cubes
- There is no perfect tiling of cubes.
- Consider the smallest cube S on the bottom.
- It cannot touch another side (see figure below,
left). - Thus S must be surrounded by larger cubes
(right). - The smallest cube on top of S also cannot touch a
side. - There are only finitely many cubes. QED
S
S
bottom view
18Perfect Tilings of Trapezoids
- There are also perfect tilings known for some
trapezoids. (Friedman, Reid, 2002)
- Open Problem Which trapezoids have perfect
tilings?
19Perfect Tilings with Small Order
- Some shapes exist that have perfect tilings of
order 2 or 3.
20The Golden Bee
- This shape also has order 2. (Scherer, 1987)
- It is called the golden bee, since r2 f and
it is in the shape of a b.
- Open Problem What other shapes have perfect
tilings?
- Open Problem What about 3-D?
21Partridge Tilings
22Partridge Tilings of Squares
- 1(1)2 2(2)2 . . . n(n)2 n(n1)/2 2.
- This means 1 square of side 1, 2 squares of side
2, up to n squares of side n have the same total
area as a square of side n(n1)/2. - If these smaller squares can be packed into the
larger square, it is called a partridge tiling. - The smallest value of ngt1 that works is called
the partridge number.
23Partridge Tilings of Squares
What is the partridge
number of a square?
a) pi b) 6 c) 8 d) 12 e) 36
24Partridge Tilings of Squares
- The first solution found was n12. (Wainwright,
1994) - The partridge number of a square is 8, and there
are 2332 solutions. (Cutler, 1996)
25Partridge Tilings of Squares
- Also solutions for 8 lt n lt 34.
- Open Problem solutions for all values of n?
- By stretching, there are partridge tilings of all
rectangles.
26Partridge Tilings of Rectangles
- A 2x1 rectangle has partridge number 7. (Cutler,
1996)
27Partridge Tilings of Rectangles
- A 3x1 rectangle has partridge number 6. (Cutler,
1996)
- A 4x1 rectangle has partridge number 7. (Hamlyn,
2001)
28Partridge Tilings of Rectangles
- A 3x2 rectangle and a 4x3 rectangle both have
partridge number 7. (Hamlyn, 2001)
- Open Problem What other rectangles have
partridge number lt 8 ?
29Partridge Tilings of Triangles
What is the partridge
number of an equilateral
triangle?
a) 7 b) 9 c) 11 d) 21 e) infinity
30Partridge Tilings of Triangles
- Equilateral triangles have partridge number 9.
(Cutler, 1996)
- By shearing, all triangles have partridge number
at most 9.
31Partridge Tilings of Triangles
What is the partridge
number of a 30-60-90 right
triangle?
a) 4 b) 5 c) 6 d) 7 e) 8
32Partridge Tilings of Triangles
- 45-45-90 triangles have partridge number 8.
(Hamlyn, 2002)
- 30-60-90 triangles have partridge number 4!
(Hamlyn, 2002)
- Open Problem What other triangles have partridge
number lt 9 ?
33Partridge Tilings of Trapezoids
- A trapezoid made from 3 equilateral triangles has
partridge number 5. (Hamlyn, 2002)
- A trapezoid made from 3/4 of a square has
partridge number 6. (Friedman, 2002)
34Partridge Tilings of Other Shapes
- A trapezoid with bases 3 and 6 and height 8 has
partridge number 4! (Reid, 1999)
- Open Problem Does any non-convex shape have a
partridge tiling?
- Open Problem Does any shape have partridge
number 2, 3, or more than 9 ?
35Reptiles and Irreptiles
36Reptiles
- A reptile is a shape that can be tiled with
smaller congruent copies of itself. - The order of a reptile is the smallest number of
congruent tiles needed to tile.
- Parallelograms and triangles are reptiles of
order (no more than) 4.
37Other Reptiles of Order 4
- Open Problem What other shapes, besides linear
transformations of these, are reptiles of order 4?
38Polyomino Reptiles
39Polyomino Reptiles
Which one of the
following shapes is a
reptile?
a) b) c) d)
e)
40Polyomino Reptiles (Reid, 1997)
41Polyiamond Reptiles (Reid, 1997)
42Reptiles
- Open Problem Which shapes are reptiles?
- Open Problem What is the order of a given
reptile? - Open Problem Are there polyomino reptiles which
cannot tile a square? - Open Problem What about 3-D?
43Reptiles
Is there a shape that is not
a reptile that can be tiled with
similar (not necessarily congruent)
copies of itself?
44Irreptiles
- An irreptile is a shape that can be tiled with
similar copies of itself. - All reptiles are irreptiles, but not all
irreptiles are reptiles, like the shape below.
45Polyomino Irreptiles(Reid, 1997)
46Trapezoid Irreptiles(Scherer, 1987)
47Irreptiles
Which one of the
following shapes is NOT an
irreptile?
Which two of these
shapes have order 5?
a) b) c) d)
e)
48Other Irreptiles(Scherer, 1987)
49Irreptiles
- Open Problem Which shapes are irreptiles?
- Open Problem What is the order of a given shape?
- Open Problem Which orders are possible?
- Open Problem What about 3-D?
50References
1 Second Book of Mathematical Puzzles
Diversions, Martin Gardner, 1961 2
Dissections of pq Rectangles, Charles Jepsen,
1996 3 Tiling with Similar Polyominoes, Mike
Reid, 2000 4 A Puzzling Journey to the
Reptiles and Related Animals, Karl Scherer,
1987 5 Packing a Partridge in a Square Tree
II, III, and IV, Robert Wainwright, 1994,
1996, 1998
51Internet References
1 http//www.meden.demon.co.uk/Fractals/golden.h
tml 2 http//clarkjag.idx.com.au/PolyPages/Repti
les.htm 3 http//mathworld.wolfram.com/PerfectSq
uareDissection.html 4 http//www.stetson.edu/ef
riedma/mathmagic/0802.html 5 http//www.math.uwa
terloo.ca/navigation/ideas/articles/
honsberger2/index.shtml 6 http//www.gamepuzzles
.com/friedman.htm