Reptiles, Partridges, and Golden Bees: Tiling Shapes with Similar Copies Erich Friedman Stetson University February 21, 2003

Perfect Tilings

Tiling Rectangles with 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)

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 Rectangles with Unequal Squares

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 Rectangles with Unequal Squares

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?

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)

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

Perfect Square Tilings Are there perfect tilings of all rectangles? The number of perfect squares of a given order: order number Open Problem: How many perfect squares of order 27?

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.

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?

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.

Perfect Tilings of Triangles Do all triangles have perfect tilings?

Perfect Tilings of Triangles There are perfect tilings for most triangles, into either 6 or 8 smaller triangles.

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

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 view S S

There are also perfect tilings known for some trapezoids. (Friedman, Reid, 2002) Open Problem: Which trapezoids have perfect tilings? Perfect Tilings of Trapezoids

And there is one more…. Perfect Tilings with Small Order Some shapes exist that have perfect tilings of order 2 or 3.

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 r 2 =  and it is in the shape of a “b”. Open Problem: What about 3-D?

Partridge Tilings

Partridge Tilings of Squares 1(1) 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.

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

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)

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.

Partridge Tilings of Rectangles A 2x1 rectangle has partridge number 7. (Cutler, 1996)

Partridge Tilings of Rectangles A 3x1 rectangle has partridge number 6. (Cutler, 1996) A 4x1 rectangle has partridge number 7. (Hamlyn, 2001)

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 ?

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

Partridge Tilings of Triangles Equilateral triangles have partridge number 9. (Cutler, 1996) By shearing, all triangles have partridge number at most 9.

Partridge Tilings of Triangles What is the partridge number of a right triangle? a) 4 b) 5 c) 6 d) 7 e) 8

triangles have partridge number 4! (Hamlyn, 2002) Partridge Tilings of Triangles triangles have partridge number 8. (Hamlyn, 2002) Open Problem: What other triangles have partridge number < 9 ?

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)

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 ?

Reptiles and Irreptiles

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.

Other Reptiles of Order 4 Open Problem: What other shapes, besides linear transformations of these, are reptiles of order 4?

Polyomino Reptiles

Which one of the following shapes is a reptile? a) b) c) d) e)

Polyomino Reptiles (Reid, 1997)

Polyiamond Reptiles (Reid, 1997)

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?

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

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.

Polyomino Irreptiles (Reid, 1997)

Trapezoid Irreptiles (Scherer, 1987)

Irreptiles Which one of the following shapes is NOT an irreptile? a) b) c) d) e) Which two of these shapes have order 5?

Other Irreptiles (Scherer, 1987)

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?

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

Internet References [1] [2] [3] [4] [5] honsberger2/index.shtml [6]