tag:blogger.com,1999:blog-72318075349108130592024-03-04T22:38:14.043-08:00Joe Bartholomewjbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.comBlogger208125tag:blogger.com,1999:blog-7231807534910813059.post-53010275612915616952014-12-29T14:22:00.000-08:002014-12-30T08:07:32.684-08:00Compound Fourier Series Surfaces<div dir="ltr" style="text-align: left;" trbidi="on">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5zFUwg7IJUs5kdZFXnbRIgYZRIYE0mpc00X3EJO1jryaue5VjyO_j07r97OQRhKFGKh5rR_tzB3-ObRoM3mO9sA8n6FDrXLhBb5Ih61UOgfSnWE7oF81IduH8rv5Q1kYDbpSGrolaQCI/s1600/Fourier_D_02.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5zFUwg7IJUs5kdZFXnbRIgYZRIYE0mpc00X3EJO1jryaue5VjyO_j07r97OQRhKFGKh5rR_tzB3-ObRoM3mO9sA8n6FDrXLhBb5Ih61UOgfSnWE7oF81IduH8rv5Q1kYDbpSGrolaQCI/s1600/Fourier_D_02.gif" height="400" width="400" /></a></div>
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<span style="font-family: Arial,Helvetica,sans-serif;">This gif demonstrates how combinations of Fourier series mathematical functions can be used to create complex animated surfaces like these: </span><span style="font-family: Arial,Helvetica,sans-serif;"><a href="http://www.joebartholomew.com/Fourier/fourier01.html">Fourier Series</a></span><span style="font-family: Arial,Helvetica,sans-serif;">, </span><span style="font-family: Arial,Helvetica,sans-serif;"><a href="http://www.joebartholomew.com/Rectify/rectify01.html">& Rectify</a></span><span style="font-family: Arial,Helvetica,sans-serif;">. Fourier series functions create certain periodic waves such as square, triangle, sawtooth, or semicircle waves from the sum of simple sine waves. In this example a <a href="https://www.processing.org/">Processing</a> program uses combinations of sine waves and Fourier series functions to simulate a complex undulating surface.<br /><br />Simple sine waves applied to the vertical </span><span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-family: Arial,Helvetica,sans-serif;">and horizontal </span>axes were used to generate a static base form:<br /> </span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWP2s5Ys0DLUhv8qzgRRbErUIHz3KEfHHvqGZi9G2AWpx2Kt7WSfm8RwVk5FPLaG0rgeI8rH1Yijd22aFYKcMzz-yQNS-wYoal7r86P1ZW3gwvO96hegnDZfDuWQheK488pp6nMBsj11Y/s1600/Fourier_D_02_1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWP2s5Ys0DLUhv8qzgRRbErUIHz3KEfHHvqGZi9G2AWpx2Kt7WSfm8RwVk5FPLaG0rgeI8rH1Yijd22aFYKcMzz-yQNS-wYoal7r86P1ZW3gwvO96hegnDZfDuWQheK488pp6nMBsj11Y/s1600/Fourier_D_02_1.png" height="200" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEise2IyJIxZxKvTRdLDNbzDN_U2QMZa6vAQtVwM85UyVcMVBJdlBvMJmMi9icH6n9fDHwFA5fdBl3gqSrweM7c4geYhJfvG06BlcxZcNsTSkjTkOPcdsKY_Bt8iFyIscl8UDYPKnRNf7GI/s1600/Fourier_D_02_2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEise2IyJIxZxKvTRdLDNbzDN_U2QMZa6vAQtVwM85UyVcMVBJdlBvMJmMi9icH6n9fDHwFA5fdBl3gqSrweM7c4geYhJfvG06BlcxZcNsTSkjTkOPcdsKY_Bt8iFyIscl8UDYPKnRNf7GI/s1600/Fourier_D_02_2.png" height="200" width="200" /></a></div>
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<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-family: Arial,Helvetica,sans-serif;">The forms are
projected in perspective with the amplitude of a sine wave modulating
the z axis depth in space. When the two are added together they make:</span></span><br />
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-family: Arial,Helvetica,sans-serif;"></span></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiM0UrSALr1r4Y7x_pO8xnFIEvn5evBS5vl4Sc2cF3Dt0XlKS1kcMULQBObwxHU8OgYwLNk6i10KExBLdXgCwCyPjf1Rzoz_EPvU3OZ0ZwccLm21zbVtRf6EEceZ0mR3dtr0OeiIXv2b14/s1600/Fourier_D_02_3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiM0UrSALr1r4Y7x_pO8xnFIEvn5evBS5vl4Sc2cF3Dt0XlKS1kcMULQBObwxHU8OgYwLNk6i10KExBLdXgCwCyPjf1Rzoz_EPvU3OZ0ZwccLm21zbVtRf6EEceZ0mR3dtr0OeiIXv2b14/s1600/Fourier_D_02_3.png" height="320" width="320" /></a></div>
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-family: Arial,Helvetica,sans-serif;"><br /></span></span>
<span style="font-family: Arial,Helvetica,sans-serif;">Fourier series triangle wave functions applied to the horizontal and vertical axes, and added together make:<br /> </span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1KUNNiQZwyeC1ko4HBjHHamMB9qLaw92zfOeKNokF70nwnbXn0X1KyOi0bd5i6BnZFEXoNXCqyx9BQJVoru-VNpvMJMAKQGQ4hs9O2qm5IYjeS2haB1o-g74BKhAAV9jKCUdjADSrkCs/s1600/Fourier_D_02_4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1KUNNiQZwyeC1ko4HBjHHamMB9qLaw92zfOeKNokF70nwnbXn0X1KyOi0bd5i6BnZFEXoNXCqyx9BQJVoru-VNpvMJMAKQGQ4hs9O2qm5IYjeS2haB1o-g74BKhAAV9jKCUdjADSrkCs/s1600/Fourier_D_02_4.png" height="320" width="320" /></a></div>
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<span style="font-family: Arial,Helvetica,sans-serif;">Using a different frequency there are four times as many triangle waves as there were sine waves. Combining all four functions, two sine and two triangle</span><span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-family: Arial,Helvetica,sans-serif;"> </span>Fourier series waves, makes:</span><br />
<span style="font-family: Arial,Helvetica,sans-serif;"> </span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9DPqAuI3kbeOx3iSjmBGckfk3ZYtC7YJCIJxYYafQcH_vaOtl9Z0Xb7IcGwGk2moNxsbQ-nvj5eHciMI3EV4fAkyJgbI0FfRR7PXvFFv4ByTyoziOtYeQje_DrSDtYtbT2ufzDMpTwAY/s1600/Fourier_D_02_5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9DPqAuI3kbeOx3iSjmBGckfk3ZYtC7YJCIJxYYafQcH_vaOtl9Z0Xb7IcGwGk2moNxsbQ-nvj5eHciMI3EV4fAkyJgbI0FfRR7PXvFFv4ByTyoziOtYeQje_DrSDtYtbT2ufzDMpTwAY/s1600/Fourier_D_02_5.png" /></a></div>
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<span style="font-family: Arial,Helvetica,sans-serif;">Animating the composite form by moving the triangle waves across the static sine waves makes the final gif, top.</span><br />
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<span style="font-family: Arial,Helvetica,sans-serif;">For simplification, two of the four functions that are combined in this example are just sine waves. The other two Fourier series functions are only a few sine functions, sufficient to create a suitable triangle wave.</span><br />
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<span style="font-family: Arial,Helvetica,sans-serif;">This example, as well as each surface in </span><span style="font-family: Arial,Helvetica,sans-serif;"><a href="http://www.joebartholomew.com/Fourier/fourier01.html">Fourier Series</a></span><span style="font-family: Arial,Helvetica,sans-serif;">, </span><span style="font-family: Arial,Helvetica,sans-serif;"><a href="http://www.joebartholomew.com/Rectify/rectify01.html">& Rectify</a></span><span style="font-family: Arial,Helvetica,sans-serif;"> could be used to model physical surfaces. </span></div>
jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com2tag:blogger.com,1999:blog-7231807534910813059.post-41280267477216444112014-12-27T14:27:00.001-08:002014-12-27T14:47:40.184-08:00& Rectify<div dir="ltr" style="text-align: left;" trbidi="on">
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<a href="http://www.joebartholomew.com/Rectify/rectifyr14.html"><img alt="& Rectify" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSMKTHF4gVBkhhJ5URsqn3GZ2JLbVTH6vyS5HzPyUv10k5uz-spEUDRuQ4ef1Ii40G4HIWWmfvAOEx11W-W3ZafugiM3DvGIZaH1Z6vZdTH6IlZMByfG02n2-FCtxJnsAtdrcIi9CPqTM/s1600/Fourier2_14r_Icon.png" height="320" title="& Rectify" width="320" /></a></div>
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<span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;">The </span><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><a href="http://www.joebartholomew.com/Rectify/rectify01.html" target="">& Rectify</a> </span>gifs are based on waveforms similar to Fourier series mathematical functions. Programs written in <a href="http://www.processing.org/" target="_blank">Processing</a>
generate and animate surfaces using combinations of sine and cosine waves. Wave frequencies determine the
number of forms within the frame. Wave amplitudes are applied to the z axis
placing each pixel forward or back in space. </span><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;">The phase changes with each frame,
moving some forms across the plane. </span>The functions used are similar to a Fourier series in which </span><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;">sine and cosine functions are combined to approximate square, sawtooth, or triangle waves. In each of these examples </span><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;">the </span>functions were modified to create new, irregular periodic forms. Multiple functions are applied to each frame. Some forms remain static while others move. Ultimately, functions are combined at the pixel level. In each case two gifs were created, with one using a rectified version of the static forms. These rectified versions are identical to their corresponding gifs except that pixels in negative z space are flipped to positive values. </span></div>
jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-8872714211619493672014-12-09T11:11:00.001-08:002014-12-27T14:15:06.100-08:00Fourier Series<div dir="ltr" style="text-align: left;" trbidi="on">
<div class="separator" style="clear: both; text-align: center;">
<a href="http://www.joebartholomew.com/Fourier/fourier31.html"><img alt="Fourier Series, 31" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcXPYoKj7NEwGIPEjC2MSiJwal4eaI9icLcA7X31OaxSOcWwD0_aIYQ5ZwQfjTFBjZAAVEYLxtvayhyphenhyphenAGaIsSk75aMVqLAmArblZrKEIad_gUdLqgivjBp4Hjuqupjb9Z8WVwZ8fwTTMA/s1600/Fourier31.png" height="320" title="Fourier Series, 31" width="320" /><span id="goog_316031987"></span></a><span id="goog_316031988"></span></div>
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<span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;">The </span><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><a href="http://www.joebartholomew.com/Fourier/fourier31.html" target="">Fourier Series</a> </span> are gifs based on waveforms similar to Fourier series mathematical functions. Programs written in <a href="http://www.processing.org/" target="_blank">Processing</a>
generate and animate surfaces using combinations of sine and cosine waves. A wave frequency determines the
number of forms within the frame. The wave amplitude is applied to the z axis
placing each pixel forward or back in space. </span><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;">The phase changes with each frame,
moving forms across the plane. </span>The functions used are similar to a Fourier series in which </span><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;">sine and cosine functions are combined to approximate square, sawtooth, or triangle waves. In each of these examples </span><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;">the </span>functions were modified to create new, irregular periodic forms.</span></div>
jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-37564672816857735662014-11-14T15:14:00.004-08:002014-11-14T15:23:23.185-08:00Facade<div dir="ltr" style="text-align: left;" trbidi="on">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGRE7zNMGfuXMa2oqpFycUM8EHAZEB8QnGWF5NgMji5R6bMzJe2yqpT_XolZqw-FQNKySkgxWqZ6mkVV9QUmKx1mWtY5Z4JGA90jtKPNzUEvlaOs-xbBVPb7jNzCZEThH8koU6xXmjXfU/s1600/Facade_Icon_600.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGRE7zNMGfuXMa2oqpFycUM8EHAZEB8QnGWF5NgMji5R6bMzJe2yqpT_XolZqw-FQNKySkgxWqZ6mkVV9QUmKx1mWtY5Z4JGA90jtKPNzUEvlaOs-xbBVPb7jNzCZEThH8koU6xXmjXfU/s1600/Facade_Icon_600.png" height="320" width="320" /></a></div>
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<span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><a href="http://www.joebartholomew.com/Facade/facade01.html" target="_blank">Facade</a> and <a href="http://www.joebartholomew.com/Facade2/facade201.html" target="_blank">Facade 2</a> are gifs built with composite waveforms. Each of 62 programs written in <a href="http://www.processing.org/" target="_blank">Processing</a> generates a gif using sine waves to animate images. The programs may calculate one or two waves for the horizontal and vertical axes (rows and columns of pixels). As in a sine wave, a frequency determines the number of waves within the frame. The phase changes with each frame, moving waves across the frame. The amplitude is applied to the z axis placing each pixel forward or back in space.</span></div>
jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-86286949337844099062014-11-14T12:11:00.003-08:002014-11-14T12:17:23.793-08:00Overtones<div dir="ltr" style="text-align: left;" trbidi="on">
<a href="http://www.joebartholomew.com/Overtones/overtones03.html" target="_blank"><br /></a>
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<a href="http://www.joebartholomew.com/Overtones/overtones03.html" imageanchor="1" style="margin-left: 1em; margin-right: 1em;" target="_blank"><img alt="http://www.joebartholomew.com/Overtones/overtones03.html" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRGKzsVeqcFUKqL2tR2hMmwzwe0x4upI42aYuSvSOGI8luuJEUdfLrua5wjO2exlcT1CnQO-4LxMIuz486NgUGgEKwtont-a2v_QF_KXIXllg7h9XsxGPsq-v-VS1ZSGuRwzmzhDZkMPM/s1600/overtones03.png" height="320" width="320" /></a></div>
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<span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><a href="http://www.joebartholomew.com/Overtones/overtones03.html" target="_blank">Overtones</a> are gifs built with </span><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;">composite waveforms. Each of the 31 programs written in <a href="http://www.processing.org/" target="_blank">Processing</a> generates a gif using slightly different wave formulas. In each case the program calculates two waves, one each for the horizontal and vertical axes (rows and columns of pixels). As in a sine wave, a frequency determines the number of waves within the frame. The phase changes with each frame, moving the waves generally from left to right. A complex form of a sine wave or similar waveform determines the amplitude.The amplitude is applied to the z axis, the x and y axes being fixed by the grid of pixels. An amplitude calculation is made for both the rows and the columns of pixels in the frame, and the two are added together to place each pixel forward or back in space.</span></div>
jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-40410729374021845662013-05-15T06:19:00.000-07:002013-05-15T06:19:01.390-07:00Green 010 <div dir="ltr" style="text-align: left;" trbidi="on">
<iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/videoseries?list=PLwbWEZOaNFLME1iLwxDLsSvynaQWx0ENV" width="560"></iframe><br />
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<span class="Text1bf">2D grids morphing in slow motion.</span> A<span class="Text3"> video in ten parts: <a href="http://www.youtube.com/playlist?list=PLwbWEZOaNFLME1iLwxDLsSvynaQWx0ENV" target="_self" title="Green 010">Green 010</a>.</span></div>
jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-29236855908235142992013-04-28T19:41:00.001-07:002013-04-29T14:59:44.799-07:00Transform<div dir="ltr" style="text-align: left;" trbidi="on">
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/5N0zLxC8zbw?feature=player_embedded' frameborder='0'></iframe></div>
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The Transform animations are eight short videos of 3D grids morphing in slow motion. Bitmap images are condensed into grids of cells. Each translucent, monochrome cell is moved forward or back based on its value. As one grid fades out another fades in:
<a href="https://www.youtube.com/playlist?list=PLwbWEZOaNFLPA88nzA_0DUQKSXk_ZQCV3" target="_blank">Transform</a><br />
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jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-24293909306492613812013-03-19T12:06:00.003-07:002013-03-19T12:10:18.044-07:00Oblique<div dir="ltr" style="text-align: left;" trbidi="on">
Oblique is a series of experimental videos in 17 parts. View all of them here: <a href="https://www.youtube.com/playlist?list=PLwbWEZOaNFLOeZNa1QJy94BkjDWS1nqaK&feature=view_all">Oblique</a><br />
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<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/s-YhuJJwlg4?feature=player_embedded' frameborder='0'></iframe></div>
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<a href="https://www.youtube.com/playlist?list=PLwbWEZOaNFLOeZNa1QJy94BkjDWS1nqaK&feature=view_all">Oblique</a>: Plot a curve. From points along the curve extend lines oblique to the plane of the curve. Make two or more such overlapping
objects. Rotate. </div>
jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-21803497299082685692013-03-17T17:40:00.002-07:002013-03-19T12:00:27.925-07:00Parallel<div dir="ltr" style="text-align: left;" trbidi="on">
Parallel is a series of experimental videos in 20 parts. View all of them here: <a href="http://www.youtube.com/playlist?list=PLwbWEZOaNFLPzI5-YjJ3p0d4TVidOMJkn">Parallel</a>.<br />
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<div class="separator" style="clear: both; text-align: center;">
<a href="http://www.youtube.com/playlist?list=PLwbWEZOaNFLPzI5-YjJ3p0d4TVidOMJkn"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/Tl4pBzFlaiA?feature=player_embedded' frameborder='0'></iframe></a></div>
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<div class="MsoPlainText">
<a href="http://www.youtube.com/playlist?list=PLwbWEZOaNFLPzI5-YjJ3p0d4TVidOMJkn">Parallel</a>: Plot a curve. From points along the curve extend lines
perpendicular to the plane of the curve. Make two or more such overlapping
objects. Rotate. Maintain parallel lines. In a few variations, extend lines to
a second, smaller curve, as in a cone. View from a line of sight perpendicular
to the plane of the curves.</div>
</div>
jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-24356386204957768632012-12-18T12:57:00.002-08:002012-12-18T13:00:56.368-08:00Six Degrees of Freedom<div dir="ltr" style="text-align: left;" trbidi="on">
<div style="text-align: center;">
<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/1oyhrS2Yfqg?feature=player_embedded' frameborder='0'></iframe> </div>
<br />
<div style="text-align: center;">
<h4>
<span style="font-family: Arial,Helvetica,sans-serif;">Click here to see <a href="http://www.youtube.com/watch?v=1oyhrS2Yfqg&list=PLwbWEZOaNFLOqOL65iZ9hgzfavnmGY56A">all seven videos</a>.</span></h4>
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<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: 11.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin;">Six
Degrees of Freedom is a program that generates symmetrical patterns from
perspective renderings of overlapping and rotating polyhedra. These wireframe
forms, built and animated with code, endlessly rotate in shared digital space.</span></span></div>
<div style="text-align: left;">
<span style="font-family: Arial,Helvetica,sans-serif;"><br /></span></div>
<div style="text-align: left;">
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: 11.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin;">Six Degrees of Freedom was exhibited at the gallery, <a href="http://www.chambersgallery.com/">Chambers @ 916</a>, November and December, 2012. </span></span></div>
</div>
jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-8018544793449726682012-08-29T12:41:00.004-07:002012-08-29T12:41:57.880-07:00Girih Polyhedra, Heptakaidecahedra with Elongated Hexagon<div dir="ltr" style="text-align: left;" trbidi="on">
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/8edOtlqvPjU?feature=player_embedded' frameborder='0'></iframe></div>
<br /></div>
jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-25885513383675432992012-08-27T15:04:00.000-07:002012-08-27T15:04:31.186-07:00Girih Polyhedra, Chalice Wireframe<div dir="ltr" style="text-align: left;" trbidi="on">
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/MKwsOOMw08s?feature=player_embedded' frameborder='0'></iframe></div>
</div>
jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-36806646675574328262012-03-15T19:24:00.003-07:002012-03-16T07:47:05.160-07:00Pentagonal Orthobirotunda Pattern<div dir="ltr" style="text-align: left;" trbidi="on">Girih patterns are decorative Islamic designs in which star shapes and polygons are connected with interlacing straight lines. Girih (Persian for “knot”) can be painstakingly created with a straightedge and compass. About nine hundred years ago Islamic architects developed an ingenious method for generating girih patterns simply with tiles. The girih tiles are five decorated polygons used to create girih patterns systematically without the need for complex drafting. We can extend the girih system into three dimensions by applying the concept to the faces of polyhedra.<br />
<br />
Here are twelve pentagonal orthobirotundi, with girih patterns.<br />
<br />
<div dir="ltr" style="text-align: left;" trbidi="on"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGsfJ987wfNvksMwC0pkNfY-4SOEKd0hkURDlNLBvBb85k6OceKJ8vM1TnKcviPi-H5N1Tgbbe58nTAw7U6qdYXeiwYpP3vafFUIF-K68BEpComjaw3i_eo61mT_trvcb6AR09HXy1Y58/s1600/pobrPattern1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGsfJ987wfNvksMwC0pkNfY-4SOEKd0hkURDlNLBvBb85k6OceKJ8vM1TnKcviPi-H5N1Tgbbe58nTAw7U6qdYXeiwYpP3vafFUIF-K68BEpComjaw3i_eo61mT_trvcb6AR09HXy1Y58/s320/pobrPattern1.jpg" width="320" /></a></div><br />
<div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dw-eB_LdtI6bYsTOdNqHqrKtdu2cX0PZbl1sCOvNWgVBk7Cl5-mtK6T9QFSKcmAW_j8TgvKI_PTEHRZfP295A' class='b-hbp-video b-uploaded' frameborder='0'></iframe></div><br />
</div></div>jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-82104434505308120932012-02-09T12:50:00.000-08:002012-02-10T15:22:27.152-08:00Girih Icosidodecahedron and Pentagonal Orthobirotunda<div dir="ltr" style="text-align: left;" trbidi="on">These two girih polyhedra are a 32-faced icosidodecahedron and the related pentagonal orthobirotunda. They each include 12 pentagon and 20 triangle faces. The pentagons are decorated with the girih tile pattern. A similar pattern continues across the equilateral triangles.<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEix_0y769s5s7pLhOPGXDwFNB1hF2aIKRYUNDJa3x15sCSRMxmNbHpn15nANSuSI39F8RuxVf740dnMELQUGz-D-Od5nbw4t2MJ17TXxpwRiNPkc2CKAIGRlRSyOy1B86YLDAvYF7KiSyo/s1600/icd2wf0.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEix_0y769s5s7pLhOPGXDwFNB1hF2aIKRYUNDJa3x15sCSRMxmNbHpn15nANSuSI39F8RuxVf740dnMELQUGz-D-Od5nbw4t2MJ17TXxpwRiNPkc2CKAIGRlRSyOy1B86YLDAvYF7KiSyo/s320/icd2wf0.jpg" width="320" /></a></div><div style="text-align: center;">Icosidodecahedron Wireframe Girih Pattern</div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKi0poL1PQMI1CARZHPFlZXI2EItEbY1Jn7eszDffpeatPM31nExtRsdGT09nLdM1vecGqdOlPSKBkOFNMg37BN6jGKp8z8as8pKgsLEG-dM05zn1q3XfGbWzTl7edoZjsIJXRlCZfKkY/s1600/icd2sm1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKi0poL1PQMI1CARZHPFlZXI2EItEbY1Jn7eszDffpeatPM31nExtRsdGT09nLdM1vecGqdOlPSKBkOFNMg37BN6jGKp8z8as8pKgsLEG-dM05zn1q3XfGbWzTl7edoZjsIJXRlCZfKkY/s320/icd2sm1.jpg" width="320" /></a></div><div style="text-align: center;">Icosidodecahedron with Girih Pattern</div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi85WjqwO-wC6Gf3Xm92fYaNFq8M_it39mxXcU9STn85BrsVSyHeLyIaIvjVWql3Oemp8tdCg-k4MA0MocZkW44PXh8-P5PPa_PTZveX24xTg4BU-hx_XfQ0CnWi2mJa_wznzHZQShATBk/s1600/icd2sm2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi85WjqwO-wC6Gf3Xm92fYaNFq8M_it39mxXcU9STn85BrsVSyHeLyIaIvjVWql3Oemp8tdCg-k4MA0MocZkW44PXh8-P5PPa_PTZveX24xTg4BU-hx_XfQ0CnWi2mJa_wznzHZQShATBk/s320/icd2sm2.jpg" width="320" /></a></div><div style="text-align: center;">Icosidodecahedron</div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgS32oiaGj7FnK1psAEb_EsY09LE9ycPreBWq2J2RmVF2_rZB26cS_rJwCwFhqRGLEoRzEBZJlY1nvMrGt43wmvXJAKI2rKrT2OC-2WkCK49WpL4PLw7VM3M6_zWVq8BVgmCGuYi3HNi9w/s1600/pobrwf0.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgS32oiaGj7FnK1psAEb_EsY09LE9ycPreBWq2J2RmVF2_rZB26cS_rJwCwFhqRGLEoRzEBZJlY1nvMrGt43wmvXJAKI2rKrT2OC-2WkCK49WpL4PLw7VM3M6_zWVq8BVgmCGuYi3HNi9w/s320/pobrwf0.jpg" width="320" /></a></div><div style="text-align: center;">Pentagonal Orthobirotunda Wireframe Girih Pattern</div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4JJRrqRXhf9qQjLZ7iW9M9I7yM0dLyQ63QKu_Q22_zpLT1NWASUH17GHUok8rVH3QqFmDGKIWN8mR1sfeNq7XBdiL9KnEplS4weCnNBvEqD_Ddgo6dhP6oGijl_NN3m_xL6WkYCYLsMA/s1600/pobrsm1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4JJRrqRXhf9qQjLZ7iW9M9I7yM0dLyQ63QKu_Q22_zpLT1NWASUH17GHUok8rVH3QqFmDGKIWN8mR1sfeNq7XBdiL9KnEplS4weCnNBvEqD_Ddgo6dhP6oGijl_NN3m_xL6WkYCYLsMA/s320/pobrsm1.jpg" width="320" /></a></div><div style="text-align: center;">Pentagonal Orthobirotunda with Girih Pattern</div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyT1i-xmrfCPMMrAMcBjW3VNRTyj8TuPT6OV0DkI-XD_RHwD6L2d-B8FwfA5Og2JZSg59AMnRmBPqDcKQ4Qow3hxVhs2DjLakFJ-WpoVmzdFPGmRdTIHUg9UJJCkFMTo-w4Y06oqWqvh4/s1600/pobrsm2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyT1i-xmrfCPMMrAMcBjW3VNRTyj8TuPT6OV0DkI-XD_RHwD6L2d-B8FwfA5Og2JZSg59AMnRmBPqDcKQ4Qow3hxVhs2DjLakFJ-WpoVmzdFPGmRdTIHUg9UJJCkFMTo-w4Y06oqWqvh4/s320/pobrsm2.jpg" width="320" /></a></div><div style="text-align: center;">Pentagonal Orthobirotunda </div></div>jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-88590857575584502402012-02-06T18:30:00.003-08:002012-04-03T07:15:59.737-07:00Chalice: Concave Girih Polyhedra with Decagons and Pentagons<div dir="ltr" style="text-align: left;" trbidi="on">These girih polyhedra have 32 faces. They include 2 decagon and 10 pentagon faces, as well as triangles. The decagon and pentagon are decorated with the girih tile pattern. A similar pattern continues across the triangles. The triangles are equilateral.<br />
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These polyhedra are related to an icosidodecahedron and pentagonal orthobirotunda. You can split an icosidodecahedron in half (making two pentagonal rotunda) and reattached the halves at pentagons to create the chalice. <br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglB1I-kQFpOoSHpc0qGVQIMPOyTtKoTa_56M4i7vHdK4Z0ODVlbBLw9DfqR0Wh-6quw5UkG-l7D2WW1rfirmqczJYxkk1MV5Ot1xfBiE73D9QSwMc3F8K8iS3TUy5sm82gNZIIoDHk-hg/s1600/cpdwf1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglB1I-kQFpOoSHpc0qGVQIMPOyTtKoTa_56M4i7vHdK4Z0ODVlbBLw9DfqR0Wh-6quw5UkG-l7D2WW1rfirmqczJYxkk1MV5Ot1xfBiE73D9QSwMc3F8K8iS3TUy5sm82gNZIIoDHk-hg/s320/cpdwf1.jpg" width="320" /></a></div><div style="text-align: center;">Chalice Wireframe Girih Pattern</div></div>jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-34787640636907946022012-02-04T16:01:00.003-08:002012-04-03T07:15:00.244-07:00Girih Polyhedra with Decagons and Elongated Hexagons<div dir="ltr" style="text-align: left;" trbidi="on">These girih polyhedra have 32 faces. They're related to the quasiregular polyhedron, the icosidodecahedron. They include 2 decagon and 10 elongated hexagon faces, as well as triangles. The decagon and elongated hexagon are decorated with the girih tile pattern. A similar pattern continues across the triangles.<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiI-XJYk7s_zS58mcVzmx2mSHhu2LCgomq3SZaPQpwBsFmwOs4GLdVnOeO_GAhvPAjo-J6uRM2MqDJHZMnLfrJrKXbzrqd48XZNfA-zy1M0TPQvjTTcfMb-i5U8se7L5Ejv5T82Wmk1ZPw/s1600/icdsm1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiI-XJYk7s_zS58mcVzmx2mSHhu2LCgomq3SZaPQpwBsFmwOs4GLdVnOeO_GAhvPAjo-J6uRM2MqDJHZMnLfrJrKXbzrqd48XZNfA-zy1M0TPQvjTTcfMb-i5U8se7L5Ejv5T82Wmk1ZPw/s320/icdsm1.jpg" width="320" /></a></div><div style="text-align: center;"> 32 Faces with Girih Pattern</div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjrFe2xmtPsqOe96334UOfMPoS3eiqd_rFIqrfiX3jUe6-CzGJfmyfW1FKw3frOW1GT7jACJBPoxWfrtRMGvYKzRR5poFG5VJr0HmNnPfBSpfUmOKoTQQKjSrjNqrwsg_SPol0ghYS7KI/s1600/icdsm2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjrFe2xmtPsqOe96334UOfMPoS3eiqd_rFIqrfiX3jUe6-CzGJfmyfW1FKw3frOW1GT7jACJBPoxWfrtRMGvYKzRR5poFG5VJr0HmNnPfBSpfUmOKoTQQKjSrjNqrwsg_SPol0ghYS7KI/s320/icdsm2.jpg" width="320" /></a></div><div style="text-align: center;">32 faces</div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsMatKqNBrrAhkBnZ7pZkuSlcswLjr1ANp3IBgSeTuc4QTNdhn_0N0jIQJYIm0g_VvnOea2UgXTk0fV526Q8PcbsxJGd4WuWtURY09MjgShTuSvsvzgDCC5g95sHcpiHGu2MgadkfEfvU/s1600/icdwf1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsMatKqNBrrAhkBnZ7pZkuSlcswLjr1ANp3IBgSeTuc4QTNdhn_0N0jIQJYIm0g_VvnOea2UgXTk0fV526Q8PcbsxJGd4WuWtURY09MjgShTuSvsvzgDCC5g95sHcpiHGu2MgadkfEfvU/s320/icdwf1.jpg" width="320" /></a></div><div style="text-align: center;"> 32 Faces with Wireframe Girih Pattern</div></div>jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-41003821529024428302012-01-21T07:39:00.000-08:002012-02-04T16:35:27.349-08:00Girih Polyhedra with Bow Tie Hexagon<div dir="ltr" style="text-align: left;" trbidi="on">These girih polyhedra are heptakaidecahedrons, but they include bow tie concave hexagons where my <a href="http://joebartholomew.blogspot.com/2012/01/girih-polyhedra.html">first version</a> had elongated hexagons, and the <a href="http://joebartholomew.blogspot.com/2012/01/girih-polyhedra-with-rhombus.html">second version</a> had rhombi. The pentagon and elongated hexagon are decorated with the girih tile pattern. A similar pattern continues across the triangles.<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj21zH7LRLrU5pBfrWKOSYJN7VlPNzi-ICY0KKakP3nz2zuZYvFovj0jyF7hWbdRiKywdft65H-HL9q9S3l_6p0afE3Jfr6iQnEYS9CeJJXWw-xBPtX_qww7svEvoe_mWrrdcJCoHG0Aho/s1600/hkdIIIsm1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj21zH7LRLrU5pBfrWKOSYJN7VlPNzi-ICY0KKakP3nz2zuZYvFovj0jyF7hWbdRiKywdft65H-HL9q9S3l_6p0afE3Jfr6iQnEYS9CeJJXWw-xBPtX_qww7svEvoe_mWrrdcJCoHG0Aho/s320/hkdIIIsm1.jpg" width="320" /></a></div><div style="text-align: center;"> Heptakaidecahedron with Girih Pattern</div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVZ-7pSNneiKQlA-2zLilpLS4KcmNiIaeEC8wis8wF_G6TFKhdAAO-_Zv5GVbhg9MOEaXkGNOmGSWWzxoz8rliM3z3v3xq1rAEO_aM5x9oEgNppBjycvbSAYGllLL5xq6w-oOG9VDSCqw/s1600/hkdIIIsm2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVZ-7pSNneiKQlA-2zLilpLS4KcmNiIaeEC8wis8wF_G6TFKhdAAO-_Zv5GVbhg9MOEaXkGNOmGSWWzxoz8rliM3z3v3xq1rAEO_aM5x9oEgNppBjycvbSAYGllLL5xq6w-oOG9VDSCqw/s320/hkdIIIsm2.jpg" width="320" /></a></div><div style="text-align: center;"> Heptakaidecahedron</div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEind5gY3WxHpkW-sNUA2VsfeuUJVdx_gNiv97YdS9sY6svMmNxP_CYhPsbgnD5jK5b1kiHzHyBhJ7Ke9lynqY1sj-V0WSU0ByEyxKfykkmL6oN5MI9UB-rwtAMofCT-CdOwFDkoZ6QkObk/s1600/hkdIIIwf1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEind5gY3WxHpkW-sNUA2VsfeuUJVdx_gNiv97YdS9sY6svMmNxP_CYhPsbgnD5jK5b1kiHzHyBhJ7Ke9lynqY1sj-V0WSU0ByEyxKfykkmL6oN5MI9UB-rwtAMofCT-CdOwFDkoZ6QkObk/s320/hkdIIIwf1.jpg" width="320" /></a></div><div style="text-align: center;"> Heptakaidecahedron Wireframe Girih Pattern</div></div>jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-24035769147945293192012-01-19T12:17:00.000-08:002012-02-04T16:37:36.609-08:00Girih Polyhedra with Rhombus<div dir="ltr" style="text-align: left;" trbidi="on">These girih polyhedra are heptakaidecahedrons, but they include rhombi where my <a href="http://joebartholomew.blogspot.com/2012/01/girih-polyhedra.html">first version</a> had elongated hexagons. The pentagon and rhombus are decorated with the girih tile pattern. A similar pattern continues across the triangles.<br />
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<div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilDmJiuMhFzEpkvdQmpy4NJ_RSqW1uLBvtKWXSEMviLvqufs22Zb2Cb_XFqmH3SHbKxGR51_4fxValF78-rmeSPwMidREhpp60pGlhrEhd2Eqd4-RMh-xa-83DJ7K-3NbR-FRmv6Ff_gM/s1600/hkdIIsm1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilDmJiuMhFzEpkvdQmpy4NJ_RSqW1uLBvtKWXSEMviLvqufs22Zb2Cb_XFqmH3SHbKxGR51_4fxValF78-rmeSPwMidREhpp60pGlhrEhd2Eqd4-RMh-xa-83DJ7K-3NbR-FRmv6Ff_gM/s320/hkdIIsm1.jpg" width="320" /></a></div><div style="text-align: center;">Heptakaidecahedron with Girih Pattern</div><div style="text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-nYNUmAMgwYwOQic6rC2Ei_WfWSLPB8AljT00LfcFLNGyhZvMtL8aDGMpvCCPQ8Vg_EMErV53YuJHT9MhS_ssAQ5cMWB16DYMWKv4fTgR5mVhjLShuc2HJW-ajmZ_Kr6_wk5QuBbMKY8/s1600/hkdIIsm2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-nYNUmAMgwYwOQic6rC2Ei_WfWSLPB8AljT00LfcFLNGyhZvMtL8aDGMpvCCPQ8Vg_EMErV53YuJHT9MhS_ssAQ5cMWB16DYMWKv4fTgR5mVhjLShuc2HJW-ajmZ_Kr6_wk5QuBbMKY8/s320/hkdIIsm2.jpg" width="320" /></a></div><div style="text-align: center;">Heptakaidecahedron</div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSPHHLhuZMC-WjcE0uXX-qLnXaq5foIiLplS6gAcVXClllKQ2HlHIEk8THtVEiAeZUEPd9zcV_58E05si59pGYsJuy3TvaeeGa-migRMQcgtj5HWPI0AfmIxE1E0ODwog9tc3pw2uEOeU/s1600/hkdIIwf1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSPHHLhuZMC-WjcE0uXX-qLnXaq5foIiLplS6gAcVXClllKQ2HlHIEk8THtVEiAeZUEPd9zcV_58E05si59pGYsJuy3TvaeeGa-migRMQcgtj5HWPI0AfmIxE1E0ODwog9tc3pw2uEOeU/s320/hkdIIwf1.jpg" width="320" /></a></div><div style="text-align: center;">Heptakaidecahedron Wireframe Girih Pattern</div><div style="text-align: center;"><br />
</div></div>jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-27458805248256497982012-01-18T16:39:00.000-08:002012-01-18T17:25:38.820-08:00Girih Polyhedra Pattern<div dir="ltr" style="text-align: left;" trbidi="on">This pattern is from twenty heptakaidecahedrons decorated with girih tile strapwork, and aligned side-by-side. The heptakaidecahedron is a seventeen sided, geometric solid with two girih tile pentagonal faces, five girih tile elongated hexagon faces, and ten isosceles triangles. In this pattern, the three dimensional solids are in five rows of four each heptakaidecahedrons with each alternating row rotated by 180 degrees, and viewed in perspective.<br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDEmUZxwa4buYmxQKExTNIYMrkN15TzOjccnkIXnyexdfUPi283dWRN3-sE8dbOuQTcqUQ-qqaZByccI5NuPGcGZg6puugqZhKoD6OcPphlrYqo63mvJRl4yfVZlHNBEXztqQ9l10EFyg/s1600/hkdsm3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDEmUZxwa4buYmxQKExTNIYMrkN15TzOjccnkIXnyexdfUPi283dWRN3-sE8dbOuQTcqUQ-qqaZByccI5NuPGcGZg6puugqZhKoD6OcPphlrYqo63mvJRl4yfVZlHNBEXztqQ9l10EFyg/s320/hkdsm3.jpg" width="320" /></a></div><br />
<div style="text-align: center;">These are the same heptakaidecahedrons at a different scale, and rotating.</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiX1FX6sqL6S9iUdc13sGr-39E3sXKAPvXA461VXKmE0hd-Bmhy__MQsK3eKV3x9CWGLaotwqqXU6oY-DRPd5-7R9zEbjCKQdI9X0a8HdQiYYFYFLL0IgAlJ0IaUFPNeugAeaOzhYRR_TI/s1600/hkdsm4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiX1FX6sqL6S9iUdc13sGr-39E3sXKAPvXA461VXKmE0hd-Bmhy__MQsK3eKV3x9CWGLaotwqqXU6oY-DRPd5-7R9zEbjCKQdI9X0a8HdQiYYFYFLL0IgAlJ0IaUFPNeugAeaOzhYRR_TI/s320/hkdsm4.jpg" width="320" /></a></div><br />
</div>jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-69831416001688437152012-01-14T17:29:00.000-08:002012-01-15T07:02:27.210-08:00Girih Polyhedra<div dir="ltr" style="text-align: left;" trbidi="on">These images are stills from an animation of rotating girih polyhedra. The two polyhedra shown here are a dodecahedron and a heptakaidecahedron. Both of these include faces that are from the <a href="http://en.wikipedia.org/wiki/Girih_tile">girih tile</a> set. The heptakaidecahedron is a seventeen sided geometric solid with two girih tile pentagonal faces, five girih tile elongated hexagon faces, and ten isosceles triangles. The triangles are not original girih tiles. The dodecahedron is a platonic solid, with twelve sides of regular pentagons. The girih tile faces can be decorated with girih lines to create a three dimensional girih pattern.<br />
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In one important sense applying girih patterns to polyhedra defeats the purpose of the girih pattern system. That is, girih tiles are designed so that the interior decoration lines continue across boundaries, tangentially. When the lines cross boundaries that are not in the same plane, the continuity is interrupted.<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhs4YWmeDd_AC4kBMQf8qqbICtq_h3NjoSLoXRo7DQLNewdHipUsnx6VzTSKpI7p1Z-DNXrYwSzwcsZhlhO_Z_FpQJBW50kjWe6-dXCidJfkd_PtzRDgQ6HSrdXMXE2mA9EGqRG3o53QmQ/s1600/ddsm1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhs4YWmeDd_AC4kBMQf8qqbICtq_h3NjoSLoXRo7DQLNewdHipUsnx6VzTSKpI7p1Z-DNXrYwSzwcsZhlhO_Z_FpQJBW50kjWe6-dXCidJfkd_PtzRDgQ6HSrdXMXE2mA9EGqRG3o53QmQ/s320/ddsm1.jpg" width="320" /></a></div><div style="text-align: center;">Dodecahedron with Girih Pattern</div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHRuTmYIWVCJ96SodmYGy59D3N9pW7UrSkW9qkhpQLtpDoImTzoRMFFj3DHWChgUIhGlG-PkiXsI2o7bPYan1Ys5nKGzr9MaN_XcCDDyjdlmUEk2-7B_dNW4TICRip5FMRlHIXbXfuvDo/s1600/ddsm2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHRuTmYIWVCJ96SodmYGy59D3N9pW7UrSkW9qkhpQLtpDoImTzoRMFFj3DHWChgUIhGlG-PkiXsI2o7bPYan1Ys5nKGzr9MaN_XcCDDyjdlmUEk2-7B_dNW4TICRip5FMRlHIXbXfuvDo/s320/ddsm2.jpg" width="320" /></a></div><div style="text-align: center;">Dodecahedron </div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhr4-JCmPNkBlt8wcTp4I2k4PGNjMcKGFkB1Ry31ZwJvNeXhp181nln6ucaZs7n6Mo3eb2cP0NXyAJD_bgfCNABVcFKY-lQeK521P_TrudlsvHgGqJ3J4GTuMP5q3T7iDFurehlCwVkVGw/s1600/ddwf0.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhr4-JCmPNkBlt8wcTp4I2k4PGNjMcKGFkB1Ry31ZwJvNeXhp181nln6ucaZs7n6Mo3eb2cP0NXyAJD_bgfCNABVcFKY-lQeK521P_TrudlsvHgGqJ3J4GTuMP5q3T7iDFurehlCwVkVGw/s320/ddwf0.jpg" width="320" /></a></div><div style="text-align: center;">Dodecahedron Wireframe Girih Pattern</div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6R8KMl5ujAyo9ofRQf3mseMC_VqU2VTKKGhQmEwvZRGpvs4T9ZTm2QVaeLheXcJNDwodKSVcy-NKK5sD8KjiEkCmQD1ouvkW85pD7fue81pi5hyphenhyphen67Q2lrVBwV9O2VB_JeRfVrKNWfhJA/s1600/hkdsm1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6R8KMl5ujAyo9ofRQf3mseMC_VqU2VTKKGhQmEwvZRGpvs4T9ZTm2QVaeLheXcJNDwodKSVcy-NKK5sD8KjiEkCmQD1ouvkW85pD7fue81pi5hyphenhyphen67Q2lrVBwV9O2VB_JeRfVrKNWfhJA/s320/hkdsm1.jpg" width="320" /></a></div><div style="text-align: center;">Heptakaidecahedron with Girih Pattern</div><br />
<div dir="ltr" style="text-align: left;" trbidi="on"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgD94HKRjWQn_dvciDuGEQGhY0q7Ujje5kQbzTRSFneoXqmMijd7kbpkM2aE81VaprkN_inIpV-cZO-jDnVvfA_sMnrVbiMmYzj_2tdDepXOWxkDNCInfSa97HXO9kKrF2_ZudEsFybeY/s1600/hkdsm2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgD94HKRjWQn_dvciDuGEQGhY0q7Ujje5kQbzTRSFneoXqmMijd7kbpkM2aE81VaprkN_inIpV-cZO-jDnVvfA_sMnrVbiMmYzj_2tdDepXOWxkDNCInfSa97HXO9kKrF2_ZudEsFybeY/s320/hkdsm2.jpg" width="320" /></a></div><div style="text-align: center;">Heptakaidecahedron </div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqVOMuf-BbmW1PQVGUhcTVrk28RsuzD0D0Wi2l-6xPE9UP1a2kU96CbP53qxKcLIFZzVYVbjWl9PFNaMkoQOnZCSbbn-_XSdjpczbkR6J936zVWHO9Wo6IQM4F3eAgMQwxot2_S0zUbjg/s1600/hkdwf0.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqVOMuf-BbmW1PQVGUhcTVrk28RsuzD0D0Wi2l-6xPE9UP1a2kU96CbP53qxKcLIFZzVYVbjWl9PFNaMkoQOnZCSbbn-_XSdjpczbkR6J936zVWHO9Wo6IQM4F3eAgMQwxot2_S0zUbjg/s320/hkdwf0.jpg" width="320" /></a></div><div style="text-align: center;">Heptakaidecahedron Wireframe Girih Pattern</div><div style="text-align: left;"><br />
</div></div></div>jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-34972043814647838932011-11-07T13:55:00.000-08:002011-11-07T15:50:38.656-08:00Five Polygons, Asymmetry<div dir="ltr" style="text-align: left;" trbidi="on">In Peter Schjeldahl’s review of the Metropolitan Museum’s new Islamic wing (“Old and New”, <i>The New Yorker</i>, Nov. 7, 2011), he says, “To grasp Islamic aesthetics, Westerners must upend their sense of ornamentation as a minor art.”<br />
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The first image below is constructed from five polygons — two shapes at three scales. The symmetrical border encloses asymmetrical, crisscrossing paths of the polygons, aligned edge-to-edge, with gaps. The second image, based on the first, uses a patterning technique originally developed by Islamic artisans. The polygon set is not Islamic, but patterning through the interior decoration of polygons is. The main difference between these images and Islamic patterns, besides the actual polygon shapes, is the use of scaling polygons to introduce variation, especially in density of line. Scaling polygons and the Islamic patterning technique borrowed for the second image below make a geometrically precise patterning system.<br />
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These polygons, based on interior angles divisible by π/7, are more difficult to arrange edge-to-edge without gaps than a polygon set based on π/5. An asymmetrical pattern like this required gaps. Nevertheless, the pattern has merit given that seemingly random connections link opposite symmetrical borders. The polygons are all precisely edge-to-edge, with no adjustments necessary. Polygons in the center align through as many as fifty other tiles to both borders. As vectors, each polygon edge has one of three lengths and fourteen possible directions.<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKBbkvuh72mCLUIvZvAS8ehI-hR5HHlleVuRfLozSUdVBjxGBIXoY0x-H2sGK6ZN9Mig9C0LFZuDYEDLqhqmF36dDg_yCNWn8syAkU_DRkmcDv7tll9gEA7h7QmJX3Nc7W16nhdorSl2g/s1600/hepta076t.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="280" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKBbkvuh72mCLUIvZvAS8ehI-hR5HHlleVuRfLozSUdVBjxGBIXoY0x-H2sGK6ZN9Mig9C0LFZuDYEDLqhqmF36dDg_yCNWn8syAkU_DRkmcDv7tll9gEA7h7QmJX3Nc7W16nhdorSl2g/s320/hepta076t.jpg" width="320" /></a></div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjuW-aeeaX8RSfr2mlg42e2K1ee99QygLl-7CvDJlsBM1DL9c7CEE8xnN_y-SP3FG3HdT-XIDe_I7FkofK9_c-Hh-dEGFHAq17IN7NWS51sFMrkn-P__Mf24PIJAsnNzt9q4Ezws6HLcs/s1600/hepta076g.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="280" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjuW-aeeaX8RSfr2mlg42e2K1ee99QygLl-7CvDJlsBM1DL9c7CEE8xnN_y-SP3FG3HdT-XIDe_I7FkofK9_c-Hh-dEGFHAq17IN7NWS51sFMrkn-P__Mf24PIJAsnNzt9q4Ezws6HLcs/s320/hepta076g.jpg" width="320" /></a></div><br />
</div>jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-75237467759294321952011-10-23T16:39:00.000-07:002011-10-23T18:40:59.166-07:007-fold<div dir="ltr" style="text-align: left;" trbidi="on">I have this quote, by Max Bill in his introduction to the catalogue of the Zürich Exhibition in 1947: "the goal of concrete art is to develop objects for mental use, just like people created objects for material use." If anyone has access to the entire document, please let me know. I'd love to read the whole thing.<br />
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These large 7-fold or 14-fold patterns with the girih-like polygon set are difficult to generate without gaps.<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgG1rvQo1OnJvj5G9aZzsx7JGyaXQO6c84UfI5SZqQCjRsQHFjhtQTadTRPXoeiciKqZP2H53Dqr84AQj6DkaIAaJ37EQWaBDbkn-JRD__lEgyD2g4fywl5sb6g8TQznPVQ7kYx1UYt3Bk/s1600/hepta063t.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgG1rvQo1OnJvj5G9aZzsx7JGyaXQO6c84UfI5SZqQCjRsQHFjhtQTadTRPXoeiciKqZP2H53Dqr84AQj6DkaIAaJ37EQWaBDbkn-JRD__lEgyD2g4fywl5sb6g8TQznPVQ7kYx1UYt3Bk/s320/hepta063t.jpg" width="320" /></a></div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJlfUvcBvu2QK4sYlKqWT5FYcky6RNnSfqpQmybCqKPU81W4rFc2qV4-ajVPAtvkMgNH3st0JypkcivQ5WiYI72o3MuWGcfF7sJpgg9NQqQC3oCEFVikdH1JbTZSC0VrQyrwqRd3SB6qQ/s1600/hepta063g.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJlfUvcBvu2QK4sYlKqWT5FYcky6RNnSfqpQmybCqKPU81W4rFc2qV4-ajVPAtvkMgNH3st0JypkcivQ5WiYI72o3MuWGcfF7sJpgg9NQqQC3oCEFVikdH1JbTZSC0VrQyrwqRd3SB6qQ/s320/hepta063g.jpg" width="320" /></a></div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhM9Vs23KJVFJy3fM-GK2JEpMdmyxoTI4RTPAG7bCqBqyPzZbMYA7BfxmLN6gSOCZuutc1mL1L5xvcD1Fg1BDLonhGBLUkr-gpK_lU4aag2iYuKzHEChp6GqsUqCnB6CTxjAe9EXZKwC40/s1600/hepta063a.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhM9Vs23KJVFJy3fM-GK2JEpMdmyxoTI4RTPAG7bCqBqyPzZbMYA7BfxmLN6gSOCZuutc1mL1L5xvcD1Fg1BDLonhGBLUkr-gpK_lU4aag2iYuKzHEChp6GqsUqCnB6CTxjAe9EXZKwC40/s320/hepta063a.jpg" width="320" /></a></div><br />
</div>jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-55451673555284244702011-10-16T16:18:00.000-07:002011-10-16T16:18:15.473-07:0014-fold Rotational Symmetry, Spiral<div dir="ltr" style="text-align: left;" trbidi="on">This spiral tiling with 14-fold symmetry was built from three equilateral polygons, including a heptagon. There are several ways to fill the center so that the tiling is a complete tessellation. A rhombus and the two polygons other than the heptagon can be used, though the complete tiling becomes 7-fold.<br />
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Edge-to-edge heptagons can be followed from the center outward. The tiling might extend infinitely(?)<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinPLcJdCZbsAm7i4UKU9oWk0fqcG5bl0M0_y022EnoHB4E-IM_ME6DjeqKVyKLxo4n_m2Zc3jG4HaPAl-NGotmQ_WnjLm6k3M1IfB6iyOGzzZkntzKGKTvOD9eYYLFdVA7yhK8HSfbTCs/s1600/hepta043t.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinPLcJdCZbsAm7i4UKU9oWk0fqcG5bl0M0_y022EnoHB4E-IM_ME6DjeqKVyKLxo4n_m2Zc3jG4HaPAl-NGotmQ_WnjLm6k3M1IfB6iyOGzzZkntzKGKTvOD9eYYLFdVA7yhK8HSfbTCs/s320/hepta043t.jpg" width="320" /></a></div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKH6bU03CcFxccEuZ0C6UNCXesEHoPYL9rfR-Q6WqPVeZNZ48LNMhOym4j1E3SWd9_L-Va5IEV1J6IdBv6Yv0122O41mqI8HIRTGLRvNnPiagZjX3f0hrS-Dx_h_YwrW4BGCPs3yoTqik/s1600/hepta043s.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKH6bU03CcFxccEuZ0C6UNCXesEHoPYL9rfR-Q6WqPVeZNZ48LNMhOym4j1E3SWd9_L-Va5IEV1J6IdBv6Yv0122O41mqI8HIRTGLRvNnPiagZjX3f0hrS-Dx_h_YwrW4BGCPs3yoTqik/s320/hepta043s.jpg" width="320" /></a></div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTf5ge_KyNY2c84x2cWYbsk-BYMkSg8YhMOcCUT_KWsfehb3yPgF7mkGe_10C9Kq8RUXbpoh1mB0ZriPZRBHMs5MEBmP1uKCpiVWTbJV8TjcNHMqcPYT8wVF3cl5osAiRzQj4yBVWdJiA/s1600/hepta043g.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTf5ge_KyNY2c84x2cWYbsk-BYMkSg8YhMOcCUT_KWsfehb3yPgF7mkGe_10C9Kq8RUXbpoh1mB0ZriPZRBHMs5MEBmP1uKCpiVWTbJV8TjcNHMqcPYT8wVF3cl5osAiRzQj4yBVWdJiA/s320/hepta043g.jpg" width="320" /></a></div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhC3xm5RD91MA5Mmko2qFBwhrntV9Xr3Ehmn5KnfTS3rKR3UyrPxpbw3QA0LQ3_w8R2PfbzmCzDH1F46lHAr9IfrglH6uJeTyUeLuWt7bm3SknvTNTeVnlWXXnqsvZc8ssNnzpwV2am1IY/s1600/hepta043a.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhC3xm5RD91MA5Mmko2qFBwhrntV9Xr3Ehmn5KnfTS3rKR3UyrPxpbw3QA0LQ3_w8R2PfbzmCzDH1F46lHAr9IfrglH6uJeTyUeLuWt7bm3SknvTNTeVnlWXXnqsvZc8ssNnzpwV2am1IY/s320/hepta043a.jpg" width="320" /></a></div><br />
</div>jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com1tag:blogger.com,1999:blog-7231807534910813059.post-45411944430992415502011-10-15T16:12:00.000-07:002011-10-15T16:15:27.010-07:00Girih Process Art<div dir="ltr" style="text-align: left;" trbidi="on">A potentially infinite process controls the development of these images. The process is difficult but not impossible to describe. It could be extremely boring to list all the steps required, and insufficient to help visualize the final image. Just covering a few guidelines, imagine six equilateral polygons including a heptagon and tetradecagon, all constructed with internal angles divisible by π/7. Arrange them edge-to-edge with no gaps and no overlaps. The radial pattern should have seven (or fourteen) fold symmetry, and be extendable forever. In my solution, outer rows of regular heptagons alternate with rows of equilateral hexagons and a rhombus.<br />
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</div>jbhttp://www.blogger.com/profile/00789548553614279286noreply@blogger.com0tag:blogger.com,1999:blog-7231807534910813059.post-48246644587395136152011-10-14T11:36:00.000-07:002011-10-14T12:00:10.349-07:00As a Design System<div dir="ltr" style="text-align: left;" trbidi="on"><a href="http://www.joebartholomew.com/digitalgirih.html">Girih Extended</a>, <a href="http://www.joebartholomew.com/arabesque12.html">Arabesque</a>, and <a href="http://www.joebartholomew.com/hepta10.html">Seven</a> are design systems similar to girih tiles, a medieval Islamic patterning technique. Like the original girih tiles, they facilitate the generation of complex patterns. Instead of constructing patterns line by line, we can design by tiling polygons. The systems are fast and accurate. The design below was generated in a few hours. The bottom diagram reflects the polygons that are selected from a menu and positioned edge-to-edge with another. The top and middle diagrams are generated automatically from the polygons.<br />
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These designs can be scaled from small to architectural applications. The final vector files are suitable for digital prints and processes.<br />
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Girih Extended, Arabesque, and Seven designs are reminiscent of Islamic art. The application of scaling, arabesque, and heptagon-based polygons to girih tiles shares much in common with, but extends girih pattern possibilities.<br />
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