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Dividing a square into similar but unequal pieces

Ed Pegg, Wolfram Research

Posted 2 years ago

CITE THIS NOTEBOOK: Dividing a square into similar but unequal pieces by Ed Pegg. Wolfram Community DEC 28 2022. Divide a square into 4 similar but unequal pieces. At an Alex Bellos column, he posted what he called a “rubbish sketch”. So, what’s b in this? Turns out its the real root of 3 x =8x+9 . In[]:= RootReduce[Solve[2b+3==Sqrt[b](b+2)][[1]]] Out[]= b 3.28 …  We can make a map of this and pick out the points we need: In[]:= b=x/.Solve[x^3==8x+9][[1]];yy={0,1,1+b,2b+3};xx=0, b ,2b+3;all=RootReduce[Tuples[{xx,yy}]];Graphics[{Table[Text[n,all[[n]]],{n,1,12}]},ImageSize->Small] Out[]= With points {{1,6},{2,7},{5,11},{3,12}} we can render the dissection perfectly: In[]:= b=x/.Solve[x^3==8x+9][[1]];pp={0,0}, b ,1,{0,1}, b ,1+b, b ,0,{2b+3,1+b},{0,1+b},{2b+3,2b+3};Graphics[{EdgeForm[{Black,Thick}],Cyan,Rectangle[pp[[#[[1]]]],pp[[#[[2]]]]]&/@Partition[Range[8],2]}] Out[]= As shown in Shattering the Plane, once you’ve got a constrained geometric image in a particular algebraic space, everything is in that algebraic space. This construction is entirely in the space of 3 x =8x+9 . In[]:= ResourceFunction["SqrtSpace"] 3.28 … ,pp Out[]= {{{0,0,0},{0,0,0}},{{0,1,0},{1,0,0}},{{0,0,0},{1,0,0}},{{0,1,0},{1,2,1}},{{0,1,0},{0,0,0}},{{9,12,4},{1,2,1}},{{0,0,0},{1,2,1}},{{9,12,4},{9,12,4}}} We can thus draw the dissection with algebraic coordinates: In[]:= root=x/.Solve[x^3==8x+9][[1]];rootcoord={{{0,0,0},{0,0,0}},{{0,1,0},{1,0,0}},{{0,0,0},{1,0,0}},{{0,1,0},{1,2,1}},{{0,1,0},{0,0,0}},{{9,12,4},{1,2,1}},{{0,0,0},{1,2,1}},{{9,12,4},{9,12,4}}};pp=ResourceFunction["SqrtSpace"][root,rootcoord];Graphics[{EdgeForm[{Black,Thick}],Cyan,Rectangle[pp[[#[[1]]]],pp[[#[[2]]]]]&/@Partition[Range[8],2]}] Out[]= George Sicherman also pointed out this dissection: Are there other solutions for dividing a square into similar but unequal pieces? For 3 pieces, we can use the plastic constant ρ: In[]:= ρ=x/.Solve[x^3==x+1][[1]];rr={{0,0},{ -2 ρ ,1},{ -2 ρ ,0},{1, -5 ρ },{ -2 ρ , -5 ρ },{1,1}};Graphics[{EdgeForm[{Black,Thick}],Green,Rectangle[rr[[#[[1]]]],rr[[#[[2]]]]]&/@Partition[Range[6],2]}] Out[]= How many ways can a square be divided into 3 similar but unequal pieces? How many ways can a square be divided into 4 similar but unequal pieces? How many ways can a square be divided into 5 similar but unequal pieces? I’m not sure this is a well explored question, so I though I’d ask it.

POSTED BY: Ed Pegg

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EDITORIAL BOARD, WOLFRAM

Posted 2 years ago

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Ed Pegg, Wolfram Research

Posted 2 years ago

John Baez, Dec 22, 2022: Dividing a square into Similar Rectangles. Covers only the case of rectangles.

POSTED BY: Ed Pegg

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