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Happy Pythagorean Day 3^2 + 4^2 = 5^2

Posted 20 days ago

Happy Pythagorean Day  3^2 + 4^2 = 5^2

9/16/25
$3^2 / 4^2 / 5^2$
$3^2 + 4^2 = 5^2$
Today's date is a Pythagorean triple, a very rare event.

We have many Pythagorean Demonstrations.

Below is a standard image ... as a puzzle, divide it into 56 identical triangles.

s = {{3, 4, 5}, {6, 8, 10}, {5, 12, 13}, {9, 12, 15}, {8, 15, 
    17}, {12, 16, 20}, {7, 24, 25}, {15, 20, 25}, {10, 24, 26}, {20, 
    21, 29}};
makeSquares[{xval_, yval_}, col_, bound_] := {Thickness[.001], col, 
   EdgeForm[{Thickness[.001], Black}], 
   Rectangle[{0, 0} + {xval, yval}, {bound, bound} + {xval, yval}], 
   Black, Table[
    Line[{{i, 0} + {xval, yval}, {i, bound} + {xval, yval}}], {i, 1, 
     bound - 1}], 
   Table[Line[{{0, i} + {xval, yval}, {bound, i} + {xval, yval}}], {i,
      1, bound - 1}]};

Manipulate[
 Graphics[{Blend[{RGBColor[.12, .61, .78], RGBColor[.67, .75, .15]}], 
   Polygon[{{0, 0}, {s[[a, 2]], 0}, {s[[a, 2]], s[[a, 1]]}, {0, 0}}], 
   makeSquares[{0, -s[[a, 2]]}, RGBColor[.67, .75, .15], s[[a, 2]]], 
   makeSquares[{s[[a, 2]], 0}, RGBColor[.12, .61, .78], s[[a, 1]]], 
   Rotate[{Thickness[.001], EdgeForm[{Thickness[.001], Black}], 
     makeSquares[{0, 0}, RGBColor[.67, .75, .15], s[[a, 3]]], 
     makeSquares[{0, 0}, RGBColor[.12, .61, .78], s[[a, 1]]]}, 
    ArcTan[s[[a, 1]]/s[[a, 2]]], {0, 0}]}, ImageSize -> {400, 400}, 
  PlotLabel -> 
   Grid[{{""}, {Style[
       Row[{Superscript[s[[a, 1]], 2], " + ", 
         Superscript[s[[a, 2]], 2], " = ", Superscript[s[[a, 3]], 2]}],
        Bold]}}]], {{a, 1, "triple"}, 1, Length[s], 1}, 
 SaveDefinitions -> True]

enter image description here

POSTED BY: Ed Pegg
2 Replies

Code for the top image.

pts={{-(12/5),41/5},{-(8/5),44/5},{-(6/5),33/5},{-1,8},{-(2/5),36/5},{0,0},{0,1},{0,2},{0,3},{0,4},{0,5},{0,10},{1/5,32/5},{3/5,46/5},{4/5,28/5},{4/5,53/5},{6/5,42/5},{7/5,24/5},{9/5,38/5},{2,0},{2,1},{2,2},{2,3},{2,4},{2,9},{12/5,34/5},{3,0},{3,1},{3,2},{3,3},{3,4},{3,5},{3,6},{16/5,37/5},{19/5,33/5},{4,8},{23/5,36/5},{5,0},{5,1},{5,2},{5,3},{5,4},{5,5},{27/5,39/5},{28/5,46/5},{29/5,28/5},{31/5,42/5},{33/5,31/5},{37/5,34/5}};

edge={{1,2},{1,5},{1,11},{2,12},{2,14},{2,15},{3,5},{3,15},{4,14},{4,17},{5,17},{5,19},{6,11},{6,20},{7,20},{7,21},{8,21},{8,22},{9,22},{9,23},{10,23},{10,24},{11,15},{11,24},{12,16},{12,26},{13,19},{13,26},{15,24},{15,26},{15,33},{16,17},{16,34},{17,25},{18,24},{18,33},{20,24},{20,27},{20,29},{22,29},{22,31},{24,31},{24,33},{25,26},{26,33},{26,45},{27,33},{27,38},{28,38},{28,39},{29,39},{29,40},{30,40},{30,41},{31,41},{31,42},{32,42},{32,43},{33,36},{33,43},{33,47},{35,43},{35,46},{36,37},{37,45},{37,46},{37,48},{38,43},{43,49},{44,48},{44,49},{45,47},{47,49}};Graphics[{

EdgeForm[None],{Cyan, Polygon[{{0,5},{16/5,37/5},{5,5}}], Red,Polygon[{{16/5,37/5},{28/5,46/5},{37/5,34/5},{5,5}}],

Green,Polygon[{{16/5,37/5},{4/5,53/5},{-(12/5),41/5},{0,5}}],Blue,Polygon[{{0,5},{0,0},{5,0},{5,5}}]},

Thick,Gray,

Line[pts[[#]]]&/@edge}]
POSTED BY: Ed Pegg

enter image description here -- you have earned Featured Contributor Badge enter image description here Your exceptional post has been selected for our editorial column Staff Picks http://wolfr.am/StaffPicks and Your Profile is now distinguished by a Featured Contributor Badge and is displayed on the Featured Contributor Board. Thank you!

POSTED BY: EDITORIAL BOARD
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