div class=ts-pagebutton class=gotoPage data-page=1Page 1button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=1 data-page=1 class=ts-thumb lazyload alt=Page 1: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails1jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=2Page 2button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=2 data-page=2 class=ts-thumb lazyload alt=Page 2: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails2jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=3Page 3button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=3 data-page=3 class=ts-thumb lazyload alt=Page 3: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails3jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=4Page 4button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=4 data-page=4 class=ts-thumb lazyload alt=Page 4: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails4jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=5Page 5button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=5 data-page=5 class=ts-thumb lazyload alt=Page 5: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails5jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=6Page 6button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=6 data-page=6 class=ts-thumb lazyload alt=Page 6: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails6jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=7Page 7button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=7 data-page=7 class=ts-thumb lazyload alt=Page 7: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails7jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=8Page 8button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=8 data-page=8 class=ts-thumb lazyload alt=Page 8: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails8jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=9Page 9button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=9 data-page=9 class=ts-thumb lazyload alt=Page 9: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails9jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=10Page 10button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=10 data-page=10 class=ts-thumb lazyload alt=Page 10: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails10jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=11Page 11button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=11 data-page=11 class=ts-thumb lazyload alt=Page 11: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails11jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=12Page 12button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=12 data-page=12 class=ts-thumb lazyload alt=Page 12: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails12jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=13Page 13button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=13 data-page=13 class=ts-thumb lazyload alt=Page 13: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails13jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=14Page 14button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=14 data-page=14 class=ts-thumb lazyload alt=Page 14: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails14jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=15Page 15button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=15 data-page=15 class=ts-thumb lazyload alt=Page 15: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails15jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=16Page 16button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=16 data-page=16 class=ts-thumb lazyload alt=Page 16: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails16jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=17Page 17button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=17 data-page=17 class=ts-thumb lazyload alt=Page 17: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails17jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=18Page 18button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=18 data-page=18 class=ts-thumb lazyload alt=Page 18: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails18jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=19Page 19button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=19 data-page=19 class=ts-thumb lazyload alt=Page 19: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails19jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=20Page 20button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=20 data-page=20 class=ts-thumb lazyload alt=Page 20: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails20jpg width=140 height=200 divdivdiv class=ts-pagebutton class=gotoPage data-page=21Page 21button div class=ts-imageimg data-url=webma-aperiodic-tilings-ergodic-theory-and-rotations-charles-radin-departmenthtmlpage=21 data-page=21 class=ts-thumb lazyload alt=Page 21: webmautexasedu · APERIODIC TILINGS ERGODIC THEORY AND ROTATIONS CHARLES RADIN Department of Mathematics University of Texas Austin TX 78712-1082 USA Abstract We use ergodic loading=lazy src=data:imagegifbase64iVBORw0KGgoAAAANSUhEUgAAAAIAAAACCAQAAADYv8WvAAAAD0lEQVR42mP8X8AwAgiABKBAv+vAXklAAAAAElFTkSuQmCC data-src=https:reader033vdocumentsmxreader033viewer20220422215ec7700b0b24422ec456110dhtml5thumbnails21jpg width=140 height=200 divdiv