CITE THIS NOTEBOOK: Discovery of pentagon tilings: celebrating Marjorie Rice centenary by Ed Pegg. Wolfram Community FEB 16 2023.
“I thought, my, that must be wonderful that someone could discover these things which no one had seen before, these beautiful patterns.” -- Marjorie Rice.
On February 16, 1923, Marjorie Rice was born. Both Wikipedia and Quanta list some of her background. Her life took a turn when she read Gardner's July 1975 Mathematical Games column, "On Tessellating the Plane with Convex Polygon Tiles". What was in that column? First, there were pentagons of types 1 to 5, found by Karl Reinhardt, though most of these were well known, for example the Cairo tiling. Karl finished up his enumeration with this statement: "Furthermore, there is a certain probability that other pentagon types, apart from those mentioned it Theorem 2, will not occur."
In[]:=
offset[type_,AA_,BB_,b_,c_,e_]:=;motif[type_,AA_,BB_,b_,c_,e_]:=;PentagonTiling[type_,{AA_,BB_,b_,c_,e_},{v1_,v2_},color_,range_,size_]:=Module[{pentset,off,pents},pentset=motif[type,AA,BB,b,c,e];off=offset[type,AA,BB,b,c,e];pents=Flatten[Table[pentset+Table[xoff[[1]]+yoff[[2]],{Length[pentset]},{5}],{x,-v1,v1},{y,-v2,v2}],2];Graphics[{EdgeForm[Thin],color,Polygon/@pents},PlotRange{{-range,range},{-range,range}},ImageSize{size,size}]];
With that, we can show the five types catalogued by Karl Reinhardt:
Out[]=
In the next part of the column, Martin introduced three relatively new results by Richard Kershner, who wrote “The author has recently succeeded in carrying through a complete determination in the special case of convex polygons. This paper contains a few critical proofs and a complete statement of the results, but not a complete proof for the excellent reason that a complete proof would require a rather large book.”
Out[]=
The column was soon challenged by Richard James III, who wrote: “Do you agree that Kershner missed this one?”
Out[]=
But it was Marjorie Rice that really made history.
Marjorie Rice: “I became fascinated by the subject and wanted to understand what made each type unique. Lacking a mathematical background, I developed my own notation system and in a few months discovered a new type which I sent to Martin Gardner. He sent it on to Doris Schattschneider to determine if it truly was a new type, and indeed it was. It was because of her interest and encouragement and the information which she sent me that I continued on in the search that eventually led to the discovery (over a two-year period) of three other new types.”
These are the four pentagon types found by Marjorie Rice:
In[]:=
Row[{PentagonTiling[9,{0,0,1.57,0,0},{5,5},RGBColor[.2,.3,.7,.5],6,150],PentagonTiling[11,{2.1,0,0,0,0},{5,5},RGBColor[.1,.8,.2,.5],3,150],PentagonTiling[12,{2.2,0,0,0,0},{5,5},RGBColor[.6,.2,.5,.5],5,150],PentagonTiling[13,{2.4,0,0,0,0},{5,5},RGBColor[.1,.9,.9,.5],5,150]},Spacer[10]]
Out[]=
Since then, two more were found, type 14 by Rolf Stein in 1985 and type 15 in 2015 by Casey Mann, Jennifer McLoud and David Von Derau.
In[]:=
Row[{PentagonTiling[14,{0,0,0,0,0},{5,5},Orange,5,150],PentagonTiling[15,{0,0,0,0,0},{5,5},Magenta,2,150]},Spacer[10]]
Out[]=
One hundred years after her birth, we thought people might like to see the discoveries of Marjorie Rice.