Reptiles, Partridges, and Golden Bees: Tiling Shapes with Similar Copies

1
Reptiles, Partridges, and Golden BeesTiling
Shapes with Similar Copies

• Erich Friedman
• Stetson University
• February 21, 2003
• efriedma_at_stetson.edu

2
Perfect Tilings
3
Tiling 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)

4
Tiling 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.)

5
Tiling Rectangleswith Unequal Squares

• bae
• cbg
• def
• fhgi

• ade
• befg
• dfh
• cgi

• Normalize with e1

6
Tiling Rectangleswith Unequal Squares
7
Perfect 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?
8
Perfect 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)

9
Perfect Square Tilings

• But eventually the smallest order of a perfect
  square tiling was shown to be 21. (Duijvestijn,
  1978)

10
Perfect 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?
11
Perfect 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)

12
Perfect 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?

13
New 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.

14
Perfect Tilings of Triangles
Do all triangles
have perfect tilings?
15
Perfect Tilings of Triangles

• There are perfect tilings for most triangles,
  into either 6 or 8 smaller triangles.

16
Perfect 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

17
Perfect 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
18
Perfect Tilings of Trapezoids

• There are also perfect tilings known for some
  trapezoids. (Friedman, Reid, 2002)

• Open Problem Which trapezoids have perfect
  tilings?

19
Perfect Tilings with Small Order

• Some shapes exist that have perfect tilings of
  order 2 or 3.

• And there is one more.

20
The 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?

21
Partridge Tilings
22
Partridge 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.

23
Partridge Tilings of Squares
What is the partridge
number of a square?
a) pi b) 6 c) 8 d) 12 e) 36
24
Partridge 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)

25
Partridge 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.

26
Partridge Tilings of Rectangles

• A 2x1 rectangle has partridge number 7. (Cutler,
  1996)

27
Partridge Tilings of Rectangles

• A 3x1 rectangle has partridge number 6. (Cutler,
  1996)

• A 4x1 rectangle has partridge number 7. (Hamlyn,
  2001)

28
Partridge 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 ?

29
Partridge Tilings of Triangles
What is the partridge
number of an equilateral
triangle?
a) 7 b) 9 c) 11 d) 21 e) infinity
30
Partridge Tilings of Triangles

• Equilateral triangles have partridge number 9.
  (Cutler, 1996)

• By shearing, all triangles have partridge number
  at most 9.

31
Partridge 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
32
Partridge 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 ?

33
Partridge 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)

34
Partridge 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 ?

35
Reptiles and Irreptiles
36
Reptiles

• 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.

37
Other Reptiles of Order 4

• Open Problem What other shapes, besides linear
  transformations of these, are reptiles of order 4?

38
Polyomino Reptiles
39
Polyomino Reptiles
Which one of the
following shapes is a
reptile?
a) b) c) d)
e)
40
Polyomino Reptiles (Reid, 1997)
41
Polyiamond Reptiles (Reid, 1997)
42
Reptiles

• 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?

43
Reptiles
Is there a shape that is not
a reptile that can be tiled with
similar (not necessarily congruent)
copies of itself?
44
Irreptiles

• 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.

45
Polyomino Irreptiles(Reid, 1997)
46
Trapezoid Irreptiles(Scherer, 1987)
47
Irreptiles
Which one of the
following shapes is NOT an
irreptile?
Which two of these
shapes have order 5?
a) b) c) d)
e)
48
Other Irreptiles(Scherer, 1987)
49
Irreptiles

• 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?

50
References
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
51
Internet 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