Randomness and Topological Invariants in Pentagonal Tiling Spaces

<table>The superposition of <svg height="19.887501" id="M250" style="vertical-align:-3.27605pt" version="1.1" viewbox="0 0 51.650002 19.887501" width="51.650002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,15.738)"><path d="M488 658v-28q-47 -3 -58.5 -12t-11.5 -39q49 -2 97 -15t93.5 -39.5t73.5 -75t28 -112.5q0 -53 -19.5 -96.5t-50 -71t-70 -47t-77.5 -27.5t-75 -9q0 -36 11.5 -45.5t58.5 -12.5v-28h-223v28q47 3 58.5 12.5t11.5 45.5q-47 1 -94.5 14t-93 39.5t-74.5 76t-29 114.5
q0 51 19 93.5t49 70t69.5 47t77.5 28t76 10.5q0 30 -11 39t-54 12v28h218zM335 118v429q-34 -2 -67 -13.5t-63.5 -35t-49.5 -65.5t-19 -96q0 -56 18.5 -99t49 -67.5t63.5 -37.5t68 -15zM418 547v-429q35 2 67.5 14t63.5 35.5t50 66.5t19 99t-19 99t-50 66.5t-64 35t-67 13.5
z" id="x3A6"></path></g>
<g transform="matrix(.012,-0,0,-.012,12.875,7.587)"><path d="M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z" id="x34"></path></g><g transform="matrix(.012,-0,0,-.012,12.875,19.825)"><path d="M619 482q0 -69 -40.5 -119t-96 -72.5t-118.5 -26.5h-44l-70 20l-31 -151q-14 -67 0.5 -83t89.5 -22l-5 -28h-287l8 28q65 7 81.5 22t29.5 83l79 398q12 56 0.5 70.5t-78.5 20.5l7 28h255q108 0 164 -43t56 -125zM524 478q0 141 -146 141q-25 0 -47 -8q-16 -6 -20.5 -13.5
t-10.5 -39.5l-43 -241q37 -13 83 -13q67 0 125.5 45t58.5 129z" id="x1D443"></path></g>
<g transform="matrix(.017,-0,0,-.017,21.075,15.738)"><path d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" id="x5B"></path></g><g transform="matrix(.017,-0,0,-.017,26.94,15.738)"><path d="M716 28l-8 -28h-254l8 28q55 5 66 17.5t5 47.5l-30 145h-232l-74 -138q-21 -43 -11 -54.5t72 -17.5l-6 -28h-235l8 28q53 7 73.5 21t53.5 70l319 540l33 8q6 -43 33 -176l80 -381q10 -49 26.5 -63t72.5 -19zM495 281l-49 264h-2l-149 -264h200z" id="x1D434"></path></g>
<g transform="matrix(.012,-0,0,-.012,39.4,19.825)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
<g transform="matrix(.017,-0,0,-.017,45.725,15.738)"><path d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" id="x5D"></path></g>
</svg> and <svg height="19.924999" id="M251" style="vertical-align:-3.27605pt" version="1.1" viewbox="0 0 51.650002 19.924999" width="51.650002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path d="M488 658v-28q-47 -3 -58.5 -12t-11.5 -39q49 -2 97 -15t93.5 -39.5t73.5 -75t28 -112.5q0 -53 -19.5 -96.5t-50 -71t-70 -47t-77.5 -27.5t-75 -9q0 -36 11.5 -45.5t58.5 -12.5v-28h-223v28q47 3 58.5 12.5t11.5 45.5q-47 1 -94.5 14t-93 39.5t-74.5 76t-29 114.5
q0 51 19 93.5t49 70t69.5 47t77.5 28t76 10.5q0 30 -11 39t-54 12v28h218zM335 118v429q-34 -2 -67 -13.5t-63.5 -35t-49.5 -65.5t-19 -96q0 -56 18.5 -99t49 -67.5t63.5 -37.5t68 -15zM418 547v-429q35 2 67.5 14t63.5 35.5t50 66.5t19 99t-19 99t-50 66.5t-64 35t-67 13.5
z" id="x3A6"></path></g>
<g transform="matrix(.012,-0,0,-.012,12.875,7.613)"><path d="M249 635q70 0 116 -43t46 -105q0 -46 -28 -80q-22 -25 -80 -64q62 -35 97 -75t35 -99q0 -81 -63 -131t-135 -50q-83 0 -137.5 45.5t-54.5 123.5q0 52 45 95q29 28 89 64q-109 62 -109 155q0 66 50.5 115t128.5 49zM238 603q-42 0 -67.5 -31t-25.5 -72q0 -50 32.5 -79.5
t98.5 -62.5q61 48 61 124q0 59 -29.5 90t-69.5 31zM248 20q46 0 76.5 33.5t30.5 89.5q0 50 -39 85.5t-110 71.5q-81 -54 -81 -137q0 -67 35.5 -105t87.5 -38z" id="x38"></path></g><g transform="matrix(.012,-0,0,-.012,12.875,19.863)"><path d="M619 482q0 -69 -40.5 -119t-96 -72.5t-118.5 -26.5h-44l-70 20l-31 -151q-14 -67 0.5 -83t89.5 -22l-5 -28h-287l8 28q65 7 81.5 22t29.5 83l79 398q12 56 0.5 70.5t-78.5 20.5l7 28h255q108 0 164 -43t56 -125zM524 478q0 141 -146 141q-25 0 -47 -8q-16 -6 -20.5 -13.5
t-10.5 -39.5l-43 -241q37 -13 83 -13q67 0 125.5 45t58.5 129z" id="x1D443"></path></g>
<g transform="matrix(.017,-0,0,-.017,21.075,15.775)"><path d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" id="x5B"></path></g><g transform="matrix(.017,-0,0,-.017,26.94,15.775)"><path d="M716 28l-8 -28h-254l8 28q55 5 66 17.5t5 47.5l-30 145h-232l-74 -138q-21 -43 -11 -54.5t72 -17.5l-6 -28h-235l8 28q53 7 73.5 21t53.5 70l319 540l33 8q6 -43 33 -176l80 -381q10 -49 26.5 -63t72.5 -19zM495 281l-49 264h-2l-149 -264h200z" id="x1D434"></path></g>
<g transform="matrix(.012,-0,0,-.012,39.4,19.863)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
<g transform="matrix(.017,-0,0,-.017,45.725,15.775)"><path d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" id="x5D"></path></g>
</svg>.</table>

Discrete Dynamics in Nature and Society

Randomness and Topological Invariants in Pentagonal Tiling Spaces

Figure 9