On 29 July, I saw a tweet that a 15th type of tiling pentagon had been discovered by Casey Mann, Jennifer McLoud-Mann, and David Von Derau at the University of Washington, Bothell. I immediately added it to my Pentagon Tilings Demonstration, then went over to Wikipedia's pentagonal tiling page to update my 2009 image made from the demo. A week later image was widely published in such places as The Guardian and npr.org.
The pentagon tiles with a motif of 12 pentagons.
pent15 ={
{{-17-6 s,2-s},{-17-6 s,6-s},{-13-6 s,3 (2+s)},{-11-6 s,6+s},{-13-6 s,2-s}},
{{-13-4 s,-s},{-11-4 s,s},{-11-4 s,8+s},{-11-6 s,6+s},{-13-6 s,2-s}},
{{-11-6 s,6+s},{-11-4 s,8+s},{-7-4 s,8+5 s},{-11-4 s,8+5 s},{-13-6 s,3 (2+s)}},
{{-13-4 s,-s},{-13-2 s,-2-s},{-5-2 s,-2-s},{-7-2 s,-2+s},{-11-4 s,s}},
{{-11-4 s,4+s},{-7-4 s,4+s},{-7,s},{-7-2 s,-2+s},{-11-4 s,s}},
{{-7-2 s,-2+s},{-7,s},{1,s},{-1,-s},{-5-2 s,-2-s}},
{{17+6 s,-2+s},{17+6 s,-6+s},{13+6 s,-3 (2+s)},{11+6 s,-6-s},{13+6 s,-2+s}},
{{13+4 s,s},{11+4 s,-s},{11+4 s,-8-s},{11+6 s,-6-s},{13+6 s,-2+s}},
{{11+6 s,-6-s},{11+4 s,-8-s},{7+4 s,-8-5 s},{11+4 s,-8-5 s},{13+6 s,-3 (2+s)}},
{{13+4 s,s},{13+2 s,2+s},{5+2 s,2+s},{7+2 s,2-s},{11+4 s,-s}},
{{11+4 s,-4-s},{7+4 s,-4-s},{7,-s},{7+2 s,2-s},{11+4 s,-s}},
{{7+2 s,2-s},{7,-s},{-1,-s},{1,s},{5+2 s,2+s}}}/8/.s->Sqrt[3];
offset15 = {{9 + 4 Sqrt[3], -8 - 5 Sqrt[3]}/4, {3 + Sqrt[3], 1 + Sqrt[3]}/4};
Here's some code to display it.
motif = Polygon /@ pent15;
colors = {{Red, Blue, Cyan}, {Yellow, Green, Purple}, {Orange, Brown, Gray}};
Graphics[{EdgeForm[Black], Table[{colors[[Mod[x, 3, 1], Mod[y, 3, 1]]], Translate[motif, x offset15[[1]] + y offset15[[2]]]},
{x, 0, 2}, {y, -5, 8}]}, PlotRange -> {{-3, 15}, {-6, 0}}, ImageSize -> {660, 220}]
With all the interest in pentagons, I figured I should make a Demonstration for tetratoid pentagons, where twelve irregular pentagons can make a polyhedron. But I haven't found the intersection between tetratoid pentagons and tiling pentagons.
What sort of image can you make from these or other pentagons?