a perspective on mathematics, the pattern, and the abstractTumblr (3.0; @intothecontinuum)http://intothecontinuum.tumblr.com/<iframe src="http://player.vimeo.com/video/67670424" width="400" height="225" frameborder="0"></iframe><br/><br/><p><a class="tumblr_blog" href="http://nldmut.tumblr.com/post/52207598745/yllogique-aardvark-from-julie-yllogique-on" target="_blank">nldmut</a>:</p> <blockquote> <p><a class="tumblr_blog" href="http://yllogique.tumblr.com/post/52163323172/aardvark-from-julie-yllogique-on-vimeo-animal-walk" target="_blank">yllogique</a>:</p> <blockquote> <p><a href="http://vimeo.com/67670424" target="_blank">Aardvark from </a><a href="http://vimeo.com/yllogique" target="_blank">Julie Yllogique</a> on <a href="http://vimeo.com" target="_blank">Vimeo</a>.</p> <blockquote> <p class="first">Animal walk - graphical interpretation</p> <p>Sound : Camille Giraudeau<br/> Patterns : <a href="http://patternstream.tumblr.com" rel="nofollow" target="_blank">patternstream.tumblr.com</a></p> </blockquote> </blockquote> <p>By Julie Yllogique (on tumblr <a href="http://yllogique.tumblr.com/" target="_blank">here</a> and <a href="http://yllogical.tumblr.com/" target="_blank">here</a>)</p> <p>featuring projected animations based off some intothecontinuum GIFs:</p> <p><a href="http://intothecontinuum.tumblr.com/post/31088724158/mathematica-code-g-x-y-z-s-p-q-r" target="_blank">1</a>, <a href="http://intothecontinuum.tumblr.com/post/36779708229/mathematica-code-rot80-table-table" target="_blank">2</a>, <a href="http://intothecontinuum.tumblr.com/post/30556308777/there-is-something-special-about-rotations-of" target="_blank">3</a>, <a href="http://intothecontinuum.tumblr.com/post/26104875937/mathematica-code-g-t-h-o-s-p" target="_blank">4</a>  and maybe <a href="http://intothecontinuum.tumblr.com/post/39634391414/inspired-by-bridget-riley-descending-1965" target="_blank">5,</a> + some from <a href="http://patternstream.tumblr.com/" target="_blank">patternstream</a></p> </blockquote>http://intothecontinuum.tumblr.com/post/52263104205http://intothecontinuum.tumblr.com/post/52263104205Wed, 05 Jun 2013 18:00:51 -0700videoJulie Yllogique<img src="http://25.media.tumblr.com/93472fc98ac702d675f2c861fdb19796/tumblr_mnhi6dRnZI1qhgxmlo1_500.gif"/><br/><br/><p><a class="tumblr_blog" href="http://ryansalge.tumblr.com/post/51521485399/a-collaboration-between-myself-and" target="_blank">ryansalge</a>:</p> <blockquote> <p>A collaboration between myself and intothecontinuum who created all the moving parts you see in the gif. You can check out his tumblr <a href="http://intothecontinuum.tumblr.com/" target="_blank">here</a>.</p> </blockquote> <p>in collaboration with <a href="http://ryansalge.tumblr.com/" target="_blank">Ryan Salge</a>!</p> <p>read more for code</p> <p><!-- more --></p> <p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre>RyanID = <br/>ImageData[<br/>ImageResize[<br/>Import["Ryan.jpg"],<br/> 500],<br/> DataReversed -> True]<br/><br/>B[x_, y_, r_, c_, o_] :=<br/>{GrayLevel[c], Opacity[o], Disk[{x, y}, r]}<br/><br/>rr[Q_] := (SeedRandom[Q]; RandomReal[])<br/><br/>Beam[IS_, f_, a_, w_, b_, o_, c_, T_, M_, t_, g_, R_, z_, zr_] :=<br/> Graphics[<br/> Table[<br/> Table[<br/> B[<br/> IS/2 + (w + a (Mod[t + s + c*rr[R*4 Q], c]))^2 (-1)^(Round[rr[Q]]) (.2 IS*rr[R*2 Q] Sin[f*2 Pi ((t + s)/c + rr[R*3 Q])]),<br/> b + Mod[t + s + c*rr[R*4 Q], c],<br/> (z + zr*rr[R*5 Q]) (.5 + .5 Abs[Sin[f*Pi ((t + s)/c + rr[R*3 Q])]]),<br/> g + 1 rr[R*6 Q],<br/> If[Mod[t + s + c*rr[R*4 Q], c] < 100, o*Mod[t + s + c*rr[R*4 Q], c]/150, o]],<br/> {s, 0, c (1 - 1/T), c/T}],<br/> {Q, 1, M, 1}],<br/> Prolog -> Raster[RyanID],<br/> PlotRange -> {{0, IS}, {0, IS*648/500}}, ImageSize -> IS]<br/><br/>Manipulate[<br/>Beam[500, 3, .0025, .6, 320, .3, 350, 10, 160, t, .2, 7, 2, 6],<br/>{t, 0, 35 - 35/8, 35/8}]</pre>http://intothecontinuum.tumblr.com/post/51528253799http://intothecontinuum.tumblr.com/post/51528253799Mon, 27 May 2013 19:38:00 -0700portraitcirclesGIFMathematicaRyan Salge<img src="http://24.media.tumblr.com/9ad5335ee49681ace07e695227618b60/tumblr_mn2murEbV51qfjvexo1_500.gif"/><br/> <br/><img src="http://24.media.tumblr.com/09e570954bd9f8aa4343c71d9cca7dd5/tumblr_mn2murEbV51qfjvexo2_500.gif"/><br/> <br/><img src="http://25.media.tumblr.com/b6e1f7972d266514748bd0d8e62f60d7/tumblr_mn2murEbV51qfjvexo3_500.gif"/><br/> <br/><p>An even number of (at least 8) regular <a href="http://en.wikipedia.org/wiki/Tetrahedron" target="_blank">tetrahedra</a> can be connected along their edges to form a ring in a way that allows them to be continuously rotated “inside-out” without disconnecting. Such configurations are commonly referred to as kaleidocycles. Shown above are kaleidocycles with 8, 10, and 12 tetrahedra exhibiting 4, 5, and 6-fold rotational symmetry, respectively. There has to be at least 8 regular tetrahedra, because any less would result in the tetrahedra colliding into each other at certain instances of the rotation. You can even make your own paper model using <a href="http://foldplay.com/kaleidocycle.action" target="_blank">this guide</a>.</p> <p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre>v1[t_] := <br/>{Cos[t], 0, Sin[t]}<br/><br/>v2[t_, a_] := <br/> 1/Sqrt[1 + Sin[t]^2 Tan[a]^2] {-Sin[t], -Sin[t] Tan[a], Cos[t]}<br/><br/>v3[t_, a_] := <br/> 1/Sqrt[1 + Sin[t]^2 Tan[a]^2] {-Sin[t]^2 Tan[a], 1, Cos[t] Sin[t] Tan[a]}<br/><br/>P[t_, a_] := <br/>{v3[t, a][[2]]/Tan[a] - v3[t, a][[1]], 0, -v3[t, a][[3]]/2}<br/><br/>Q[t_, a_] := <br/>{v3[t, a][[2]]/Tan[a], v3[t, a][[2]], v3[t, a][[3]]/2}<br/><br/>vertices[t_, a_] := <br/>{P[t, a] - Sqrt[2]/2 v1[t], P[t, a] + Sqrt[2]/2 v1[t],<br/> Q[t, a] - Sqrt[2]/2 v2[t, a], Q[t, a] + Sqrt[2]/2 v2[t, a]}<br/><br/>Tetrahedron[T_, t_, a_, o_] :=<br/> Table[<br/> {FaceForm[White], Opacity[o], EdgeForm[Thick],<br/> Polygon[<br/> Table[<br/> T[vertices[t, a][[1 + Mod[i + j, 4]]]], {i, 1, 3, 1}]]},<br/> {j, 0, 3, 1}]<br/><br/>Kaleidocycle[pr_, t_, n_, o_, A_] := Graphics3D[<br/> Rotate[<br/> Table[<br/> Rotate[<br/> Table[<br/> Tetrahedron[T, t, 2 Pi/n, o],<br/> {T, {TransformationFunction[IdentityMatrix[4]], <br/> ReflectionTransform[{-Sin[2 Pi/n], Cos[2 Pi/n], 0}]}}],<br/> r*4 Pi/n, {0, 0, 1}],<br/> {r, 0, n - 1, 1}],<br/> A*Sin[t], {0, 1, 0}],<br/> PlotRange -> pr, ImageSize -> 500, Axes -> False, Boxed -> False, <br/> Lighting -> "Neutral", ViewPoint -> {0, 0, 2}, Background -> White ]<br/><br/>Manipulate[<br/>Kaleidocycle[pr, t, n, o, A],<br/> {pr, 1.5, 50}, {t, 0, 2 Pi}, {n, 8, 16, 1},{o, 1, 0}, {{A, 0}, 0, 2 Pi}]</pre>http://intothecontinuum.tumblr.com/post/50873970770http://intothecontinuum.tumblr.com/post/50873970770Sun, 19 May 2013 19:07:00 -0700GIFMathematicakaleidocycle<img src="http://25.media.tumblr.com/4be8160470f71bea55bc5dab850dfe40/tumblr_mmb7i9VYqc1qfjvexo1_500.gif"/><br/><br/><div class="post_content clearfix" id="post_content_48274888385"> <div class="post_text_wrapper"> <p>Made in response to a recent question posted at the <a href="http://mathematica.stackexchange.com/questions/24148/how-can-this-image-optical-illusion-be-created-with-mathematica" target="_blank"> Mathematica Stack Exchange</a>.</p> <p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre><code><span class="pln">img </span><span class="pun">=</span><span class="pln"> </span><span class="kwd">ImageCrop</span><span class="pun">@</span><span class="kwd">DensityPlot</span><span class="tag">[</span><span class="pln"> </span><span class="kwd">Sin</span><span class="tag">[</span><span class="lit">2</span><span class="pln"> x </span><span class="pun">-</span><span class="pln"> </span><span class="lit">20</span><span class="pln"> </span><span class="kwd">Log</span><span class="tag">[</span><span class="lit">2</span><span class="pln"> </span><span class="tag">(</span><span class="kwd">Sin</span><span class="tag">[</span><span class="pln">y</span><span class="tag">]</span><span class="pun">^</span><span class="lit">2</span><span class="pln"> </span><span class="pun">+</span><span class="pln"> </span><span class="lit">1</span><span class="tag">)</span><span class="pun">,</span><span class="pln"> </span><span class="lit">2</span><span class="tag">]]</span><span class="pun">,</span><span class="pln"> </span><span class="tag">{</span><span class="pln">x</span><span class="pun">,</span><span class="pln"> </span><span class="lit">0</span><span class="pun">,</span><span class="pln"> </span><span class="lit">16</span><span class="pln"> </span><span class="kwd">Pi</span><span class="tag">}</span><span class="pun">,</span><span class="pln"> </span><span class="tag">{</span><span class="pln">y</span><span class="pun">,</span><span class="pln"> </span><span class="lit">0</span><span class="pun">,</span><span class="pln"> </span><span class="lit">32</span><span class="pln"> </span><span class="kwd">Pi</span><span class="tag">}</span><span class="pun">,</span><span class="pln"> </span><span class="kwd">PlotPoints</span><span class="pln"> </span><span class="pun">-></span><span class="pln"> </span><span class="lit">250</span><span class="pun">,</span><span class="pln"> </span><span class="kwd">ColorFunction</span><span class="pln"> </span><span class="pun">-></span><span class="pln"> </span><span class="str">"SunsetColors"</span><span class="pun">,</span><span class="pln"> </span><span class="kwd">Frame</span><span class="pln"> </span><span class="pun">-></span><span class="pln"> </span><span class="kwd">False</span><span class="pun">,</span><span class="pln"> </span><span class="kwd">ImageSize</span><span class="pln"> </span><span class="pun">-></span><span class="pln"> </span><span class="lit">600</span><span class="tag">]</span></code><br/><code><span class="pln"><br/>LogPolar</span><span class="tag">[</span><span class="atv">x_</span><span class="pun">,</span><span class="pln"> </span><span class="atv">y_</span><span class="tag">]</span><span class="pln"> </span><span class="pun">:=</span><span class="pln"> </span><span class="tag">{</span><span class="kwd">Log</span><span class="tag">[</span><span class="kwd">Sqrt</span><span class="tag">[</span><span class="pln">x</span><span class="pun">^</span><span class="lit">2</span><span class="pln"> </span><span class="pun">+</span><span class="pln"> y</span><span class="pun">^</span><span class="lit">2</span><span class="tag">]]</span><span class="pun">,</span><span class="pln"> </span><span class="kwd">ArcTan</span><span class="tag">[</span><span class="pln">x</span><span class="pun">,</span><span class="pln"> y</span><span class="tag">]}<br/><br/></span></code><code><span class="pln">d </span><span class="pun">=</span><span class="pln"> </span><span class="kwd">ImageDimensions</span><span class="tag">[</span><span class="pln">img</span><span class="tag">][[</span><span class="lit">1</span><span class="tag">]]</span><span class="pln"> <br/>Manipulate</span><span class="tag">[</span><span class="pln"> </span><span class="kwd">ImageResize</span><span class="tag">[</span><span class="pln"> </span><span class="kwd">ImageTransformation</span><span class="tag">[</span><span class="pln"> </span><span class="kwd">ImageTake</span><span class="tag">[</span><span class="pln"> img</span><span class="pun">,</span><span class="pln"> </span><span class="tag">{</span><span class="lit">1</span><span class="pun">,</span><span class="pln"> </span><span class="lit">14</span><span class="pun">*</span><span class="pln">d</span><span class="pun">/</span><span class="lit">16</span><span class="tag">}</span><span class="pun">,</span><span class="pln"> </span><span class="tag">{</span><span class="lit">1</span><span class="pln"> </span><span class="pun">+</span><span class="pln"> </span><span class="tag">(</span><span class="lit">2</span><span class="pln"> </span><span class="pun">-</span><span class="pln"> </span><span class="lit">2</span><span class="pln"> t</span><span class="tag">)</span><span class="pun">*</span><span class="pln">d</span><span class="pun">/</span><span class="lit">32</span><span class="pun">,</span><span class="pln"> </span><span class="tag">(</span><span class="lit">32</span><span class="pln"> </span><span class="pun">-</span><span class="pln"> </span><span class="lit">2</span><span class="pln"> t</span><span class="tag">)</span><span class="pun">*</span><span class="pln">d</span><span class="pun">/</span><span class="lit">32</span><span class="tag">}]</span><span class="pun">,</span><span class="pln"> LogPolar</span><span class="tag">[</span><span class="atv">#</span><span class="tag">[[</span><span class="lit">1</span><span class="tag">]]</span><span class="pun">,</span><span class="pln"> </span><span class="atv">#</span><span class="tag">[[</span><span class="lit">2</span><span class="tag">]]]</span><span class="pln"> </span><span class="pun">&,</span><span class="pln"> </span><span class="kwd">DataRange</span><span class="pln"> </span><span class="pun">-></span><span class="pln"> </span><span class="tag">{{</span><span class="pun">-</span><span class="kwd">Pi</span><span class="pun">,</span><span class="pln"> </span><span class="kwd">Pi</span><span class="tag">}</span><span class="pun">,</span><span class="pln"> </span><span class="tag">{</span><span class="pun">-</span><span class="kwd">Pi</span><span class="pun">,</span><span class="pln"> </span><span class="kwd">Pi</span><span class="tag">}}]</span><span class="pun">,</span><span class="pln"> </span><span class="lit">500</span><span class="tag">]</span><span class="pun">,</span><span class="pln"> </span><span class="tag">{</span><span class="pln">t</span><span class="pun">,</span><span class="pln"> </span><span class="lit">0</span><span class="pun">,</span><span class="pln"> </span><span class="lit">6</span><span class="pun">/</span><span class="lit">7</span><span class="pun">,</span><span class="pln"> </span><span class="lit">1</span><span class="pun">/</span><span class="lit">7</span><span class="tag">}]</span><span class="pln"></span></code></pre> </div> </div>http://intothecontinuum.tumblr.com/post/49735694481http://intothecontinuum.tumblr.com/post/49735694481Sun, 05 May 2013 18:00:44 -0700logpolar coordinatesform constantsdensity plotGIFMathematica<img src="http://24.media.tumblr.com/fc9b9d47fe285620b55b237834d63d08/tumblr_mlgbscgSMu1qfjvexo1_500.gif"/><br/><br/><p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre>ID = ImageData[<br/>Binarize[Rasterize[<br/>Import["Erwin.jpg"], <br/>RasterSize -> 100], .7], <br/>DataReversed -> True]<br/><br/>Tile[k_, rx_, ry_, x_, y_, r_] :=<br/>Table[<br/> Translate[<br/> Rotate[<br/> {AbsoluteThickness[k],<br/> Circle[{i, i}, {rx, ry}, {i*Pi, Pi/2 + i*Pi}]},<br/> r, {.5, .5}],<br/> {x, y}],<br/> {i, 0, 1, 1}]<br/><br/>Parquet[IS_, X_, Y_, u_, v_, k_, rx1_, rx2_, ry1_, ry2_, f_, t_] :=<br/>Graphics[<br/> Table[<br/> Tile[<br/> k,<br/> rx1 + rx2*x/X,<br/> ry1 + ry2/2*(1 + Sin[2 Pi (f*y/Y - t)]),<br/> x, y, <br/> ID[[1 + v*y, 1 + u*x]] Pi/2],<br/> {x, 3, X, 1}, {y, 5, Y, 1}],<br/> ImageSize -> IS, PlotRange -> {{3, X + 1}, {5, Y + 1}}]<br/><br/>Manipulate[<br/>Parquet[500, 56, 60, 2, 2, 2, .5, 0, 0, .5, .5, t],<br/>{t, 0, 17/18, 1/18}]</pre>http://intothecontinuum.tumblr.com/post/48320145595http://intothecontinuum.tumblr.com/post/48320145595Thu, 18 Apr 2013 18:01:07 -0700GIFErwin Schrödingerparquet deformationportraitImage processingMathematica<img src="http://25.media.tumblr.com/091f05a32a4d33a35888515d14be8328/tumblr_mk8v5qxfZe1qfjvexo4_r1_500.gif"/><br/> <br/><img src="http://24.media.tumblr.com/bf994f4769819676dabae3452e5d126c/tumblr_mk8v5qxfZe1qfjvexo1_500.gif"/><br/> <br/><img src="http://24.media.tumblr.com/bb2112b317c70a14a809c8db54081ce3/tumblr_mk8v5qxfZe1qfjvexo2_r1_500.gif"/><br/> <br/><img src="http://25.media.tumblr.com/e0f21bf65237a0f1338387010d30b9cd/tumblr_mk8v5qxfZe1qfjvexo5_r1_500.gif"/><br/> <br/><img src="http://25.media.tumblr.com/8e36e0735a306d8f3e577c64548ac455/tumblr_mk8v5qxfZe1qfjvexo3_r1_500.gif"/><br/> <br/><p>In a <a href="http://intothecontinuum.tumblr.com/post/46211876653/consider-tiling-the-plane-using-only-square-tiles" target="_blank">previous post</a> we experienced what a <a href="http://en.wikipedia.org/wiki/Truchet_tiles" target="_blank">Truchet tiling</a> looks like. This time, the animations above show what it might be like to look down on a Truchet tiling, but <em>while moving</em> along it in a straight line.</p> <p>In each of the animations the tiling is really only shifting in one direction (the direction corresponding to “up” when viewed on your screen) as made apparent in the following simple Truchet tiling:</p> <p><img alt="" src="http://media.tumblr.com/94b44aaabd9090eab31ee946b015506c/tumblr_inline_mk8vfaVSN11qz4rgp.gif"/></p> <p>For an increased dramatic effect, the rate of this upward movement in the animations is made to correspond with the frame rate of the GIF in such a way where precisely one row of the tiling leaves our view every frame. This makes the smoothly translating tiling just shown look like this instead:</p> <p><img alt="" src="http://media.tumblr.com/03191dd568b623e3c04b36b45f515122/tumblr_inline_mk8vfhlcQc1qz4rgp.gif"/></p> <p>This explains why the individual tiles seem to be changing orientations in place, and why there appears to be a static grid of horizontal and vertical lines outlining the tiles. Together with the geometry of the configurations in the tilings, this also explains why there seems to be motion is several different directions for any given tiling. Perhaps one might be able to realistically create a similar effect if they manage to run along such a tiling at a rate which corresponds to the <a href="http://en.wikipedia.org/wiki/Persistence_of_vision" target="_blank">“frame rate” of human vision</a>.</p> <p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre>T[r_, x_, y_] :=<br/> Translate[<br/> Rotate[<br/> {EdgeForm[Thickness[0]], Polygon[{{1, 0}, {0, 0}, {0, 1}}]},<br/> r, {.5, .5}],<br/> {x, y}]<br/><br/>Manipulate[<br/> Graphics[<br/> Table[<br/> T[Mod[a*x + b*y, m] Pi/2, x + h, y + v],<br/> {x, 1, X, 1}, {y, 1, Y + 60, 1}],<br/> PlotRange -> {{1, X}, {1, Y - .1}}, ImageSize -> 500],<br/> {{X, 23}, 1, 100, 1}, {{Y, 23}, 1, 100, 1},<br/> {{m, 11}, 1, 100, 1},<br/> {{a, 18}, 1, 100, 1},<br/> {{b, 14}, 1, 100, 1},<br/> {h, 0, 14},<br/> {v, 0, 14}]</pre>http://intothecontinuum.tumblr.com/post/46303986723http://intothecontinuum.tumblr.com/post/46303986723Mon, 25 Mar 2013 19:04:00 -0700GIFTruchet tilingtilingsMathematicaepilepsy warning<img src="http://25.media.tumblr.com/22a7b8f1d19bb49d0ee9058ce4db10a5/tumblr_mk6rcjdDj31qfjvexo2_500.jpg"/><br/> <br/><img src="http://24.media.tumblr.com/9ce2ababc164d3e2b96184d5722a8467/tumblr_mk6rcjdDj31qfjvexo3_r1_500.jpg"/><br/> <br/><img src="http://25.media.tumblr.com/38906342cb4833cfab0ca0bbff756e78/tumblr_mk6rcjdDj31qfjvexo1_r2_500.jpg"/><br/> <br/><img src="http://24.media.tumblr.com/6f5fc52839c7d1f4f9e133ba1a7eeddf/tumblr_mk6rcjdDj31qfjvexo4_500.jpg"/><br/> <br/><img src="http://25.media.tumblr.com/8df14a9ba9bc895d46a4cbc2228ccca3/tumblr_mk6rcjdDj31qfjvexo5_500.jpg"/><br/> <br/><img src="http://25.media.tumblr.com/76fc2adf8eff7011a699e781cac48267/tumblr_mk6rcjdDj31qfjvexo8_500.jpg"/><br/> <br/><img src="http://24.media.tumblr.com/bcbcd0f554ac13aaf5e200dd99d05344/tumblr_mk6rcjdDj31qfjvexo7_500.jpg"/><br/> <br/><img src="http://25.media.tumblr.com/74805d126dc3806b7fb4ca81614e0d11/tumblr_mk6rcjdDj31qfjvexo9_r1_500.jpg"/><br/> <br/><p>Consider tiling the plane using only square tiles like this:</p> <p><img alt="" src="http://media.tumblr.com/d485577824a7a2d31206bb99e6c61d98/tumblr_inline_mk6s7gxYHU1qz4rgp.jpg"/></p> <p>The 4-fold rotational symmetry of the square allows a tile to be placed in the 4 different orientations shown here:</p> <p><img alt="" src="http://media.tumblr.com/151f0244216649e96ac5c80554d365ce/tumblr_inline_mk6scbCxKh1qz4rgp.jpg"/></p> <p>Despite these constraints there are still a lot of different ways to tile the plane. Shown above are a few examples constructed with an algorithm using <a href="http://en.wikipedia.org/wiki/Modular_arithmatic" target="_blank">modular arithmetic</a>. This essentially makes the tiles along different rows follow the same sequence but shifted over by some amount.</p> <p>Each of the tilings shown are actually periodic and can tile the entire plane. <br/>(e.g. <a href="http://bon.gs/tile/#tile=http://25.media.tumblr.com/22a7b8f1d19bb49d0ee9058ce4db10a5/tumblr_mk6rcjdDj31qfjvexo2_500.jpg" target="_blank">1</a>, <a href="http://bon.gs/tile/#tile=http://24.media.tumblr.com/9ce2ababc164d3e2b96184d5722a8467/tumblr_mk6rcjdDj31qfjvexo3_r1_500.jpg" target="_blank">2</a>, <a href="http://bon.gs/tile/#tile=http://25.media.tumblr.com/38906342cb4833cfab0ca0bbff756e78/tumblr_mk6rcjdDj31qfjvexo1_r2_500.jpg" target="_blank">3</a>, <a href="http://bon.gs/tile/#tile=http://25.media.tumblr.com/74805d126dc3806b7fb4ca81614e0d11/tumblr_mk6rcjdDj31qfjvexo9_r1_500.jpg" target="_blank">4</a>)</p> <p>Tilings of this variety are called <a href="http://en.wikipedia.org/wiki/Truchet_tiles" target="_blank">Truchet tilings</a>.</p> <p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre>T[r_, x_, y_] :=<br/> Translate[<br/> Rotate[<br/> {EdgeForm[Thickness[0]], Polygon[{{1, 0}, {0, 0}, {0, 1}}]},<br/> r, {.5, .5}],<br/> {x, y}]<br/><br/>Manipulate[<br/> Graphics[<br/> Table[<br/> T[Mod[a*x + b*y, m] Pi/2, x, y],<br/> {x, 1, X, 1}, {y, 1, Y, 1}],<br/> PlotRange -> {{1, X}, {1, Y - .1}}, ImageSize -> 500],<br/> {{X, 29}, 1, 100, 1}, {{Y, 29}, 1, 100, 1},<br/> {{m, 14}, 1, 100, 1},<br/> {{a, 5}, 1, 100, 1},<br/> {{b, 5}, 1, 100, 1}]</pre>http://intothecontinuum.tumblr.com/post/46211876653http://intothecontinuum.tumblr.com/post/46211876653Sun, 24 Mar 2013 18:01:00 -0700tilingsTruchet tiling2coloringMathematica<p>Here is the icon:</p> <p><img alt="image" src="http://media.tumblr.com/e3ce61f6448f59c5b84f427f2b258c6d/tumblr_inline_mjsh87x2fy1qf3sji.png"/></p> <p>which is a crop from this photo of <a href="http://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger" target="_blank">Erwin Schrödinger</a>:</p> <p><img alt="image" src="http://media.tumblr.com/f100e87c3c3062443d3750d7e7fcbd93/tumblr_inline_mjsh9kEkq21qf3sji.jpg"/></p> <p>I do not know any other details such as who took it, where it was taken, and when. If anyone does know, please <a href="http://intothecontinuum.tumblr.com/ask" target="_blank">let me know</a>.</p>http://intothecontinuum.tumblr.com/post/45563095439http://intothecontinuum.tumblr.com/post/45563095439Sat, 16 Mar 2013 22:23:00 -0700Erwin Schrödinger<img src="http://25.media.tumblr.com/7a9ed88548fe4ae50e5ca548a8ffb424/tumblr_mjh3tdtQVy1qfjvexo1_500.gif"/><br/><br/><p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre>P[A_, f_, w_, h_, M_, Y_, t_] :=<br/> Plot[<br/> Table[<br/> A*Sin[f*x + t + n*2 Pi/w] + h*n,<br/> {n, 1, M, 1}],<br/> {x, 0, 4 Pi}, <br/> PlotStyle -> Directive[Black, AbsoluteThickness[3]], <br/> PlotRange -> {{0, 4 Pi}, {-.3, Y}}, Axes -> False,<br/> AspectRatio -> 5/7, ImageSize -> {700, 500}]<br/><br/>Manipulate[<br/> ImageRotate[<br/> P[.35, 1, 28, .6, 57, 35, t],<br/> -Pi/2],<br/> {t, 2 Pi, Pi/10, -Pi/10}]</pre>http://intothecontinuum.tumblr.com/post/45077556257http://intothecontinuum.tumblr.com/post/45077556257Sun, 10 Mar 2013 19:00:37 -0700GIFMathematicawavywaterfall plot<img src="http://25.media.tumblr.com/d7ce2cff8de0db10eacc48cedb9ee3f9/tumblr_mj3xbwjiW51qfjvexo1_500.gif"/><br/><br/><div class="caption"> <p>Inspired by <a href="http://www.vasiljgodzh.blogspot.com/" target="_blank">Vasilj Godzh</a></p> <p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre>s[q_] := (SeedRandom[q]; RandomReal[])<br/><br/>r[S_, a_, v_, w_, t_] := <br/> S (1 + .05 Sin[v*a] Cos[w*a] + .1 Cos[8*a] + .025 Sin[a + t])<br/><br/>F[Q_, S_, M_, v_, w_, th_, t_] :=<br/> {EdgeForm[{AbsoluteThickness[th], Black}], FaceForm[White],<br/> Polygon[<br/> Table[<br/> {{0, 0},<br/> {r[S, (a + s[Q*a]) 2 Pi/M, v, w, t] Cos[(a + s[Q*a]) 2 Pi/M],<br/> r[S, (a + s[Q*a]) 2 Pi/M, v, w, t] Sin[(a + s[Q*a]) 2 Pi/M]},<br/> {r[S, (a + 1 + s[Q (Mod[a, M] + 1)]) 2 Pi/M, v, w, t] Cos[(a + 1 + s[Q (Mod[a, M] + 1)]) 2 Pi/M],<br/> r[S, (a + 1 + s[Q (Mod[a, M] + 1)]) 2 Pi/M, v, w, t] Sin[(a + 1 + s[Q (Mod[a, M] + 1)]) 2 Pi/M]}},<br/> {a, 1, M, 1}]]}<br/><br/>Manipulate[<br/> Graphics[<br/> Table[<br/> Translate[<br/> Reverse@<br/> Table[<br/> F[i*j, (1 + .3 i^1.7), 125 + 25 i, <br/> 3 + Round[9 s[i*j]], 3 + Round[9 s[2 i*j]],<br/> .6, t + s[j] 2 Pi],<br/> {i, 1, 4, 1}],<br/> {17*s[j], 23.8*s[2 j]}],<br/> {j, 1, 46, 1}],<br/> PlotRange -> {{.5, 17.5}, {-1.2, 22.6}}, <br/> ImageSize -> {500, 700}],<br/> {t, 0, 2Pi}]</pre> </div>http://intothecontinuum.tumblr.com/post/44508281621http://intothecontinuum.tumblr.com/post/44508281621Sun, 03 Mar 2013 18:07:00 -0800GIFMathematicaVasilj Godzh<img src="http://25.media.tumblr.com/b79f3ebe554c228848f48a43b25bff1d/tumblr_mipzbzlB911qfjvexo1_r1_500.gif"/><br/> <br/><img src="http://24.media.tumblr.com/bae255649228ea2f920c52e4c730458c/tumblr_mipzbzlB911qfjvexo2_r1_500.gif"/><br/> <br/><img src="http://24.media.tumblr.com/d977b6ba82d3079605436ea681906ba6/tumblr_mipzbzlB911qfjvexo4_r1_500.gif"/><br/> <br/><div class="caption"> <p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre><br/>G[A_, B_, s_, N_, T_, t_, th_, pr_, u_, v_] :=<br/> Graphics[<br/> {EdgeForm[{AbsoluteThickness[th], Black}], FaceForm[White], <br/> Polygon[<br/> Table[<br/> {{0, 0},<br/> {(Cos[u*a+t]Sin[v*a+t])Cos[a+t], (Cos[u*a+t]Sin[v*a+t])Sin[a+t]},<br/> {(Cos[u*a+s+t]Sin[v*a+s+t])Cos[a+s+t], (Cos[u*a+s+t]Sin[v*a+s+t])Sin[a+s+t]}},<br/> {a, B + T, A + T, 2 Pi/N}]]},<br/> PlotRange -> {{-1.5 pr, 1.5 pr}, {-pr, pr}}, ImageSize -> 400]<br/><br/>Manipulate[<br/>G[2Pi, 0, .5, 200, 0, t, .3, .7, 1, v],<br/>{v, {3,4,5}},<br/>{t,0,2Pi}] </pre> </div>http://intothecontinuum.tumblr.com/post/44026309325http://intothecontinuum.tumblr.com/post/44026309325Mon, 25 Feb 2013 17:30:14 -0800GIFMathemtica<img src="http://25.media.tumblr.com/5cfff026fe21b80425c52ff4dbce1124/tumblr_mipbfbUCsK1qfjvexo1_500.gif"/><br/><br/><div class="caption"> <p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre><br/>G[A_, B_, s_, N_, T_, t_, pr_, th_] :=<br/> Graphics[<br/> {EdgeForm[{AbsoluteThickness[th], Black}], FaceForm[White], <br/> Polygon[<br/> Table[<br/> {{0, 0},<br/> {(Cos[4a+s*t]+Sin[4a+s*t])Cos[a+s*t], (Cos[4a+s*t]+Sin[4a+s*t]) Sin[a+s*t]},<br/> {(Cos[4a+s+s*t]+Sin[4a+s+s*t])Cos[a+s+s*t], (Cos[4a+s+s*t]+Sin[4a+s+s*t]) Sin[a+s+s*t]}},<br/> {a, B + T, A + T, 2 Pi/N}]]},<br/> PlotRange -> pr, ImageSize -> 500]<br/><br/>Manipulate[<br/>G[2Pi, 0, .75, 200, T, 0, 1.5, .3],<br/>{T,0,2Pi}] </pre> </div>http://intothecontinuum.tumblr.com/post/43941149014http://intothecontinuum.tumblr.com/post/43941149014Sun, 24 Feb 2013 16:49:00 -0800GIFMathematica<img src="http://25.media.tumblr.com/06fe8a050921832b5179b66b13ff7ed7/tumblr_mic9l0uPPP1qfjvexo1_500.gif"/><br/><br/><div class="caption"> <p>Inspired by <a href="http://nldmut.tumblr.com/post/43075180684/matam-elongated-fall" target="_blank">this<br/></a></p> <p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre>Manipulate[<br/> Graphics[<br/> Table[<br/> Table[<br/> {EdgeForm[{Black, AbsoluteThickness[2.5]}], FaceForm[White],<br/> Translate[<br/> Polygon[<br/> Table[{<br/> {3.9 Sin[Pi/8 (t + V)]<br/> + i/8 ((-1)^S*(1 + Abs[3.9 Sin[Pi/8 (t + V)]]) - 3.9 Sin[Pi/8 (t + V)]),<br/> i/8},<br/> {-3.9 Sin[Pi/8 (t + V)] <br/> + i/8 ((-1)^S*(1 + Abs[3.9 Sin[Pi/8 (t + V)]]) + 3.9 Sin[Pi/8 (t + V)]),<br/> 2 - i/8},<br/> {(-1)^S* 6 - (-1)^S*i/8, 2 - i/8},<br/> {(-1)^S*6 - (-1)^S*i/8, i/8}},<br/> {i, 0, 8, 1}]],<br/> {0, -2 V}]},<br/> {S, 0, 1}],<br/> {V, 0, 7, 1}],<br/> ImageSize -> 500],<br/> {t, 0, 16}]</pre> </div>http://intothecontinuum.tumblr.com/post/43270305784http://intothecontinuum.tumblr.com/post/43270305784Sat, 16 Feb 2013 17:02:20 -0800GIFMathematica<img src="http://25.media.tumblr.com/5a4be317a0f1bc8cc97a50094b9348b6/tumblr_mh3qgwLJag1qfjvexo1_500.gif"/><br/><br/><p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre>Tile[k_, rx_, ry_, x_, y_, r_] :=<br/> Table[<br/> Translate[<br/> Rotate[<br/> {AbsoluteThickness[k],<br/> Circle[{i, i}, {rx, ry}, {i*Pi, Pi/2 + i*Pi}]},<br/> r, {.5, .5}],<br/> {x, y}],<br/> {i, 0, 1, 1}]<br/><br/>rr[Q_] := (SeedRandom[Q]; RandomReal[])<br/><br/>GCOS[k_, rx1_, rx2_, ry1_, ry2_, X_, Y_, Q_, t_] :=<br/> Graphics[<br/> Table[<br/> Tile[k,<br/>  .75 + .25 Cos[2 Pi*Mod[rx1 + rx2*x/X + t, 1]],<br/>  .75 + .25 Cos[2 Pi*Mod[ry1 + ry2*y/Y + t, 1]],<br/>  x, y, Floor[3*rr[Q*x*y]] Pi/2],<br/> {x, 1, X, 1}, {y, 1, Y, 1}],<br/> ImageSize -> 500, PlotRange -> {{1, X + 1}, {1, Y + 1}}]<br/><br/>Manipulate[<br/> GCOS[2, .5, .5, .5, .5, 30, 42, 4, t],<br/>{t, .05, 1, .05}]</pre>http://intothecontinuum.tumblr.com/post/41325052098http://intothecontinuum.tumblr.com/post/41325052098Wed, 23 Jan 2013 18:00:00 -0800parquet deformationtiligsMathematicaGIF<img src="http://25.media.tumblr.com/48df606bb58aadbbb2d5ee2efc6350a5/tumblr_mh0nclHlKx1qfjvexo1_500.jpg"/><br/><br/><p><a href="http://25.media.tumblr.com/48df606bb58aadbbb2d5ee2efc6350a5/tumblr_mh0nclHlKx1qfjvexo1_1280.jpg" target="_blank">1000x1000</a></p> <p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre>Tile[k_, rx_, ry_, x_, y_, r_] :=<br/> Table[<br/> Translate[<br/> Rotate[<br/> {AbsoluteThickness[k],<br/> Circle[{i, i}, {rx, ry}, {i*Pi, Pi/2 + i*Pi}]},<br/> r, {.5, .5}],<br/> {x, y}],<br/> {i, 0, 1, 1}]<br/><br/>rr[Q_] := (SeedRandom[Q]; RandomReal[])<br/><br/>Manipulate[<br/>Graphics[<br/> Table[<br/> Tile[4, .5 + .5*x/1000, y/1000, x, y, Floor[3*rr[x*y]] Pi/2],<br/> {x, 1, 100, 1}, {y, 1, 100, 1}],<br/> ImageSize -> 1000, PlotRange -> {{1, 101}, {1, 101}}],<br/>{r, 0, 1, .25}]</pre>http://intothecontinuum.tumblr.com/post/41248138935http://intothecontinuum.tumblr.com/post/41248138935Tue, 22 Jan 2013 19:00:24 -0800parquet deformationtilings<img src="http://25.media.tumblr.com/61eb8c8dab516e0b3f8f010c2c664396/tumblr_mh0kw8rJZz1qfjvexo1_r1_500.jpg"/><br/><br/><p>Testing the new <a href="http://staff.tumblr.com/post/40779375054/panoramas" target="_blank">panorama feature</a> on Tumblr with a <em>parquet deformation</em>—a kind of <a href="http://www.tess-elation.co.uk/parquet-deformations---an-intro" target="_blank">“geometrical tessellating metamorphosis”</a>. Douglas Hofstadter has a bit to say about them in <a href="http://books.google.com/books?id=o8jzWF7rD6oC&lpg=PA192&ots=jQAh7zMpls&dq=parquet%20deformation&pg=PA191#v=onepage&q&f=false" target="_blank">Metamagical Themas</a>.</p> <p>Click the image to view in panorama mode. The <a href="http://i.imgur.com/JMJr0V2.jpg" target="_blank">original image</a> is 4000 pixels wide.</p> <p>more variants: <a href="http://i.imgur.com/JMJr0V2.jpg" target="_blank">1</a>, <a href="http://i.imgur.com/AY02uVT.jpg" target="_blank">2</a>, <a href="http://i.imgur.com/6Q02MxX.jpg" target="_blank">3</a></p> <p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre>Tile[k_, rx_, ry_, x_, y_, r_] :=<br/> Table[<br/> Translate[<br/> Rotate[<br/> {AbsoluteThickness[k],<br/> Circle[{i, i}, {rx, ry}, {i*Pi, Pi/2 + i*Pi}]},<br/> r, {.5, .5}],<br/> {x, y}],<br/> {i, 0, 1, 1}]<br/><br/>rr[Q_] := (SeedRandom[Q]; RandomReal[])<br/><br/>Manipulate[<br/>Graphics[<br/> Table[<br/> Tile[4, .5 + .5*x/200, r , x, y, Floor[3*rr[x*y]] Pi/2],<br/> {x, 1, 200, 1}, {y, 1, 20, 1}],<br/> ImageSize -> 4000, PlotRange -> {{1, 201}, {1, 21}}],<br/>{r, 0, 1, .25}]<br/><br/><br/></pre>http://intothecontinuum.tumblr.com/post/41242766946http://intothecontinuum.tumblr.com/post/41242766946Tue, 22 Jan 2013 18:00:31 -0800parquet deformationtilingsMathematica<img src="http://24.media.tumblr.com/8abee39617911fdb0e694e0c766fe922/tumblr_mgspenmZOx1qfjvexo1_500.gif"/><br/> <br/><img src="http://25.media.tumblr.com/02be0e75443b9b54924fc405efd36020/tumblr_mgspenmZOx1qfjvexo2_500.gif"/><br/> <br/><img src="http://24.media.tumblr.com/f3562bdd4e7b59ab5eacde9e7cafb2ae/tumblr_mgspenmZOx1qfjvexo3_500.gif"/><br/> <br/><p>Imagine throwing a bunch of balls up into the air at just the right place, at just the right time, with just the right speed, so that all the balls reach their maximum height at the same exact time and form a picture. Simulated here is the tumblr logo, a smiley face, and a more complicated face. Can you recognize who?</p> <p>Some unrealistic physical assumptions:<br/>- The balls don’t collide into each other.<br/>- The balls only bounce straight up and down.</p> <p>Inspired by <a href="http://nldmut.tumblr.com/post/40272274773/cineraria-2013-on-vimeo-this-is-blowing-my" target="_blank">this</a>.</p> <p>Read more for code:</p> <p><!-- more --></p> <p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre>ImgData[image_, rs_, b_] :=<br/> ImageData[<br/> Binarize[<br/> Rasterize[<br/> Import[image],<br/> RasterSize -> rs],<br/> b],<br/> DataReversed -> True]<br/><br/>tumblrID = ImgData[tumblr.jpg, 64, .5];<br/><br/>smileyID = ImgData[smiley.jpg, 64, .5];<br/><br/>erwinID = ImgData[erwin.jpg, 42, .76];<br/><br/>B[v_, N_] := If[N == -1, 0, Sum[2*.8^n*v, {n, 0, N, 1}]]<br/><br/>P[t_, v_, N_] := -.5 (t - B[v, N])^2 + .8^(N + 1)*v*(t - B[v, N])<br/><br/>S[y_, h_] := Sqrt[2 (y + h)]<br/><br/>Projectiles[ImageData_, PR_, Y_, dy_, dx_, r_, T_, h_, t_] := <br/> Graphics[<br/> Table[<br/> Table[<br/> Disk[<br/> {Position[ImageData[[y]], 0][[x, 1]],<br/> If[t < T - S[y, h], 0,<br/> If[0 < t - T + S[y, h] < B[S[y, h], 0], P[t - T + S[y, h], S[y, h], -1],<br/> If[B[S[y, h], 0] < t - T + S[y, h] < B[S[y, h], 1], P[t - T + S[y, h], S[y, h], 0],<br/> If[B[S[y, h], 1] < t - T + S[y, h] < B[S[y, h], 2], P[t - T + S[y, h], S[y, h], 1],<br/> If[B[S[y, h], 2] < t - T + S[y, h] < B[S[y, h], 3], P[t - T + S[y, h], S[y, h], 2],<br/> If[B[S[y, h], 3] < t - T + S[y, h] < B[S[y, h], 4], P[t - T + S[y, h], S[y, h], 3],<br/> 0]]]]]]},<br/> r],<br/> {x, 1, Length@Position[ImageData[[y]], 0], dx}],<br/> {y, 1, Y, dy}],<br/> PlotRange -> PR, ImageSize -> 500]<br/><br/>Manipulate[<br/>Projectiles[tumblrID, {{5, 620}, {-8, 300}}, 160, 12, 11, 5, 25, 100, t] ,<br/>{t, 0, 160, 2}]<br/><br/>Manipulate[<br/>Projectiles[smileyID, {{-10, 230}, {-8, 300}}, 225, 10, 10, 4.5, 25, 70, t] ,<br/>{t, 0, 170, 2.5}]<br/><br/>Manipulate[<br/>Projectiles[erwinID, {{5, 109}, {-2, 140}}, 225, 6, 5, 2, 20, 25, t] ,<br/>{t, 0, 120, 1.5}]</pre>http://intothecontinuum.tumblr.com/post/40807939205http://intothecontinuum.tumblr.com/post/40807939205Thu, 17 Jan 2013 18:01:00 -0800GIFMathematicaphysicsImage processingportraitcirclesErwin Schrödinger<img src="http://25.media.tumblr.com/c5da3a57127bd30fd0b99b9d913d7763/tumblr_mgckjwkerw1rxpsomo7_400.gif"/><br/> <br/><img src="http://24.media.tumblr.com/17c7173d0269a92baa7f0e0b38be8c5e/tumblr_mgckjwkerw1rxpsomo1_400.gif"/><br/> <br/><img src="http://25.media.tumblr.com/2772c43975ac857819e235e3815e5d87/tumblr_mgckjwkerw1rxpsomo2_400.gif"/><br/> <br/><img src="http://24.media.tumblr.com/aaad830f5ef8bb4862d87b422a590a54/tumblr_mgckjwkerw1rxpsomo3_400.gif"/><br/> <br/><img src="http://24.media.tumblr.com/0aebb02f67840dc3b1d5a38573ac2729/tumblr_mgckjwkerw1rxpsomo4_400.gif"/><br/> <br/><img src="http://24.media.tumblr.com/ef59ad9fbe737e00c5ffc0d085866373/tumblr_mgckjwkerw1rxpsomo5_400.gif"/><br/> <br/><img src="http://25.media.tumblr.com/967c17fc403924e3783c80b035a6a5f7/tumblr_mgckjwkerw1rxpsomo6_r1_400.gif"/><br/> <br/><img src="http://25.media.tumblr.com/0d24cefa2f96fd97a11a1d5672e47fd3/tumblr_mgckjwkerw1rxpsomo8_400.gif"/><br/> <br/><img src="http://24.media.tumblr.com/9b88f8b7dc183dc99f2d096b71c62de9/tumblr_mgckjwkerw1rxpsomo9_r1_400.gif"/><br/> <br/><img src="http://25.media.tumblr.com/6ee25d8e9ce3f756e4c12266231d4387/tumblr_mgckjwkerw1rxpsomo10_400.gif"/><br/> <br/><p><a class="tumblr_blog" href="http://nldmut.tumblr.com/post/40138469391/specular-holograms-by-matthew-brand-currently-on" target="_blank">nldmut</a>:</p> <blockquote> <div> <p><a href="http://en.wikipedia.org/wiki/Specular_holography" target="_blank">Specular holograms</a> by Matthew Brand currently on display at the new <a href="http://momath.org/" target="_blank">Museum of Mathematics</a> in New York.</p> <p>See <a href="http://www.zintaglio.com/knots.html" target="_blank">his site</a> for more.</p> </div> </blockquote> <p>The technique used by Brand to create these pieces is not one of conventional holography. He meticulously controls the unique shape of thousands of tiny optical pieces placed on a surface creating a 3D effect when the light source or viewer moves. This is essentially a mathematical problem in <a href="http://en.wikipedia.org/wiki/Differential_geometry" target="_blank">differential geometry</a> and <a href="http://en.wikipedia.org/wiki/Combinatorial_optimization" target="_blank">combinatorial optimization</a>. Brand was the first person to correctly describe this technique in 2008 even though it dates back as early as the 1930s (check out <a href="http://arxiv.org/abs/1101.0301" target="_blank">his paper</a> for details).</p>http://intothecontinuum.tumblr.com/post/40138589843http://intothecontinuum.tumblr.com/post/40138589843Wed, 09 Jan 2013 17:44:00 -0800GIFmathknot theoryspecular holographyMoMathMatthew Brand3Dstereoscopic<img src="http://25.media.tumblr.com/7e15031d98c34333a40207b52b8b2ef3/tumblr_mg72iz6RNY1qfjvexo1_500.gif"/><br/> s=.2<br/><br/> <img src="http://25.media.tumblr.com/495a4bb8d6174d6653c0efd4f60937e2/tumblr_mg72iz6RNY1qfjvexo2_500.gif"/><br/> s=.5<br/><br/> <img src="http://25.media.tumblr.com/02a0ca2077f0ca12b5775739327377fe/tumblr_mg72iz6RNY1qfjvexo4_500.gif"/><br/> s=.9<br/><br/> <p><a href="http://intothecontinuum.tumblr.com/tagged/logpolar-coordinates" target="_blank">log-polar transforms</a></p> <p>shown with increasing wave amplitude from top to bottom</p> <p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre>WfPlot[ s_, t_] :=<br/> Graphics[<br/> Table[<br/> {AbsoluteThickness[1.5], <br/> Line[<br/> Table[<br/> {i + If[Mod[i, 2] == 0, s*Sin[j*2 Pi/66 + i*2 Pi/6 + t], 0],<br/> (-1)^i*.5 + .4*j},<br/>  {i, 1, 19}]]},<br/> {j, 1, 69, 1}],<br/> PlotRange -> {{1, 19}, {.8, 27.2}}, <br/> ImageSize -> {500, 500}]<br/><br/>LogPolar[x_, y_] := {Log[Sqrt[x^2 + y^2]], ArcTan[x, y]}<br/><br/>Manipulate[<br/> ImageTransformation[<br/> WfPlot[s, t],<br/> LogPolar[#[[1]], #[[2]]] &, DataRange -> {{-Pi, Pi}, {-Pi, Pi}}],<br/>{s, 0, 1}, {t, 0, 2Pi}]</pre>http://intothecontinuum.tumblr.com/post/39976670322http://intothecontinuum.tumblr.com/post/39976670322Mon, 07 Jan 2013 18:00:00 -0800GIFMathematicalogpolar coordinateswavy<img src="http://24.media.tumblr.com/8b33dc09793bfd3f4b7fc9acd72afbdb/tumblr_mg71mu8Ur81qfjvexo1_500.gif"/><br/> log-polar coordinates<br/><br/> <img src="http://24.media.tumblr.com/df527e36849587e498120d6c77df6e6c/tumblr_mg71mu8Ur81qfjvexo2_500.gif"/><br/> Cartesian coordinates<br/><br/> <div class="caption"> <p>Some more inspiration from <a href="http://intothecontinuum.tumblr.com/post/39634391414/inspired-by-bridget-riley-descending-1965" target="_blank">Bridget Riley</a> — think <a href="http://www.wikipaintings.org/en/bridget-riley/blaze-1-1962" target="_blank">Blaze 1 (1962)</a>.<br/>The first image is what you get after transforming the second image into <a href="http://intothecontinuum.tumblr.com/post/19754693526/visual-hallucinations-and-form-constants" target="_blank">log-polar coordinates</a>.</p> <p><a href="http://www.wolfram.com/mathematica/" target="_blank">Mathematica</a> code:</p> <pre>WfPlot[ t_ ] :=<br/> Graphics[<br/> Table[<br/> {AbsoluteThickness[3], <br/> Line[<br/> Table[<br/> {i + If[Mod[i, 2] == 0, .5*Sin[j*2 Pi/66 + t], 0],<br/> (-1)^i*.5 + .4*j},<br/>  {i, 1, 19}]]},<br/> {j, 1, 69, 1}],<br/> PlotRange -> {{1, 19}, {.8, 27.2}}, <br/> ImageSize -> {500, 500}]<br/><br/><br/>Manipulate[<br/> WfPlot[ t ],<br/>{t, 0, 2Pi}]<br/><br/>LogPolar[x_, y_] := {Log[Sqrt[x^2 + y^2]], ArcTan[x, y]}<br/><br/>Manipulate[<br/> ImageTransformtion[<br/> WfPlot[ t ],<br/> LogPolar[#[[1]], #[[2]]] &, DataRange -> {{-Pi, Pi}, {-Pi, Pi}}],<br/>{t,0,2Pi}]</pre> </div>http://intothecontinuum.tumblr.com/post/39891322436http://intothecontinuum.tumblr.com/post/39891322436Sun, 06 Jan 2013 18:04:00 -0800GIFMathematicaBridget Rileylogpolar coordinateswaterfall plotwavy