reptiles, partridges, and golden bees: tiling shapes with similar copies erich friedman stetson...
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![Page 1: Reptiles, Partridges, and Golden Bees: Tiling Shapes with Similar Copies Erich Friedman Stetson University February 21, 2003 efriedma@stetson.edu](https://reader036.vdocuments.mx/reader036/viewer/2022062515/56649cf45503460f949c2c32/html5/thumbnails/1.jpg)
Reptiles, Partridges, and Golden Bees:
Tiling Shapes with Similar Copies
Erich FriedmanStetson UniversityFebruary 21, 2003
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Perfect Tilings
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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)
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• 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 Kirchoff’s Laws if each edge has unit resistance.)
Tiling Rectangleswith Unequal Squares
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• b=a+e• c=b+g• d=e+f• f+h=g+i
• a=d+e• b+e=f+g• d+f=h• c+g=i
• Normalize with e=1
Tiling Rectangleswith Unequal Squares
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Tiling Rectangleswith Unequal Squares
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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?
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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, 1950’s)
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Perfect Square Tilings
• But eventually the smallest order of a perfect square tiling was shown to be 21. (Duijvestijn, 1978)
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Perfect Square Tilings
Are there perfect tilings of all rectangles?
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?
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Perfect Tilings of Rectangles
• The order of a 2x1 rectangle is 8 (Jepsen, 1996)
• There are perfect tilings of all rectangles since we can stretch a perfect tiling of squares.
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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?
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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 n+m-1 copies.– Take an n-tiling of S, and replace the smallest
tile with an m-tiling of S.
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Perfect Tilings of Triangles
Do all triangles have perfect tilings?
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Perfect Tilings of Triangles
• There are perfect tilings for most triangles, into either 6 or 8 smaller triangles.
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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
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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
bottom viewS S
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• There are also perfect tilings known for some trapezoids. (Friedman, Reid, 2002)
• Open Problem: Which trapezoids have perfect tilings?
Perfect Tilings of Trapezoids
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• And there is one more….
Perfect Tilings with Small Order
• Some shapes exist that have perfect tilings of order 2 or 3.
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• This shape also has order 2. (Scherer, 1987)
• Open Problem: What other shapes have perfect tilings?
The Golden Bee
• It is called the “golden bee”, since r2 = and it is in the shape of a “b”.
• Open Problem: What about 3-D?
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Partridge Tilings
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Partridge Tilings of Squares
• 1(1)2 + 2(2)2 + . . . + n(n)2 = [ n(n+1)/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(n+1)/2.
• If these smaller squares can be packed into the larger square, it is called a partridge tiling.
• The smallest value of n>1 that works is called the partridge number.
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Partridge Tilings of Squares
What is the partridge number of a square?
a) pi b) 6 c) 8 d) 12 e) 36
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Partridge Tilings of Squares
• The first solution found was n=12. (Wainwright, 1994)
• The partridge number of a square is 8, and there are 2332 solutions. (Cutler, 1996)
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Partridge Tilings of Squares• Also solutions
for 8 < n < 34.
• Open Problem: solutions for all values of n?
• By stretching, there are partridge tilings of all rectangles.
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Partridge Tilings of Rectangles• A 2x1 rectangle has partridge
number 7. (Cutler, 1996)
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Partridge Tilings of Rectangles
• A 3x1 rectangle has partridge number 6. (Cutler, 1996)
• A 4x1 rectangle has partridge number 7. (Hamlyn, 2001)
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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 < 8 ?
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Partridge Tilings of Triangles
What is the partridge number of an equilateral triangle?
a) 7 b) 9 c) 11 d) 21 e) infinity
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Partridge Tilings of Triangles
• Equilateral triangles have partridge number 9. (Cutler, 1996)
• By shearing, all triangles have partridge number at most 9.
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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
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• 30-60-90 triangles have partridge number 4! (Hamlyn, 2002)
Partridge Tilings of Triangles
• 45-45-90 triangles have partridge number 8. (Hamlyn, 2002)
• Open Problem: What other triangles have partridge number < 9 ?
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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)
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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 ?
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Reptiles and Irreptiles
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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.
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Other Reptiles of Order 4
• Open Problem: What other shapes, besides linear transformations of these, are reptiles of order 4?
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Polyomino Reptiles
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Polyomino Reptiles
Which one of the following shapes is a reptile?
a) b) c) d) e)
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Polyomino Reptiles (Reid, 1997)
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Polyiamond Reptiles (Reid, 1997)
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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?
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Reptiles
Is there a shape that is not a reptile that can be tiled with similar (not necessarily congruent) copies of itself?
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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.
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Polyomino Irreptiles(Reid, 1997)
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Trapezoid Irreptiles(Scherer, 1987)
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Irreptiles
Which one of the following shapes is NOT an irreptile?
a) b) c) d) e)
Which two of these shapes have order 5?
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Other Irreptiles(Scherer, 1987)
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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?
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References[1] “Second Book of Mathematical Puzzles & Diversions”, Martin Gardner, 1961
[2] “Dissections of p:q 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
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Internet References[1] http://www.meden.demon.co.uk/Fractals/golden.html
[2] http://clarkjag.idx.com.au/PolyPages/Reptiles.htm
[3] http://mathworld.wolfram.com/PerfectSquareDissection.html
[4] http://www.stetson.edu/~efriedma/mathmagic/0802.html
[5] http://www.math.uwaterloo.ca/navigation/ideas/articles/ honsberger2/index.shtml
[6] http://www.gamepuzzles.com/friedman.htm