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III",storyMedia1:"myEscher1.png",storyMedia2:"myEscher2.png",storyMedia3:"myEscher3.png",storyMedia4:"",storyText:"Dutch artist M.C. Escher relied on math to create his art. These images illustrate the steps in constructing Escher's Woodcut Circle Limit III (1959). The underlying structure is that of hyperbolic tessellation (3, 4, 3, 4, 3, 4). The color and the theory of symmetry is used to find the basic tile, circled in red in the next image. Finally, the choice of the three-fish pattern (left bottom) for the tile recreates Woodcut Circle Limit III."},{title:"Soccer Ball Reinvented",imgName:"soccerBall.png",altText:"Why are soccer balls made out of pentagons and hexagons? Any alternatives for octagon lovers?",storyMedia1:"soccerBall1.png",storyMedia2:"soccerBall2.png",storyMedia3:"soccerBall3.png",storyMedia4:"soccerBall4.png",storyText:"A soccer ball is not a sphere; it’s an approximation built from pentagons and hexagons. More precisely, there are two pentagons and a hexagon at each corner, or (5, 5, 6). Spheres may be approximated many ways with polygons. For example, the polyhedron (3, 3, 3, 3, 5), consisting of four triangles and a pentagon at each vertex, is also a tessellation of a sphere. Viewing these deconstructions (a wireframe of just pentagons and just triangles), the tessellation reinvents the soccer ball."},{title:"The Shape of the Universe",imgName:"shapeUni.png",altText:"Flat or curved? What is the shape of space we live in?",storyMedia1:"shapeUni.gif",storyMedia2:"shapeUni1.png",storyMedia3:"shapeUniRot.gif",storyMedia4:"",storyText:'What would it look like to live in a so called hyperbolic space, where space curves "negatively" -- just the opposite of the "positively curved" sphere? Since the Earth is not flat, the roads we travel on are not straight lines -- they lie on the surface of our planet. We experience a very small part of our universe that appears flat, but if we could see it from a mathematical perspective, like the octagon in this animation, we would agree that our universe is curved. Einstein\'s "Theory of General Relativity" states that the curvature of space is what we experience as gravity. So, "curved space" is not just idle curiosity.'},{title:"Caught in a Dual Net",imgName:"caughtInaDualNetF.png",altText:"",storyMedia1:"caughtInaDualNetF1.png",storyMedia2:"caughtInaDualNetF2.png",storyMedia3:"caughtInaDualNetF3.png",storyMedia4:"",storyText:"In 1976, Felix Klein wrote about a tessellation with seven triangles at each vertex (3, 3, 3, 3, 3, 3, 3) shown in red. This image, titled Caught in a Dual Net (2008), contains (3, 3, 3, 3, 3, 3, 3) combined with a tessellation with three heptagons at each vertex in purple, revealing a new tessellation (6, 6, 7). The turquoise heptagon and two hexagons are bounded by wire models of (3, 3, 3, 3, 3, 3, 3) and (7, 7, 7, 7, 7, 7, 7) with edges alternating between red and purple."},{title:"Tessellation or Not?",imgName:"seaPearls.png",altText:"",storyMedia1:"seaPearls1.png",storyMedia2:"seaPearls2.png",storyMedia3:"",storyMedia4:"",storyText:"Sometimes a tessellation’s pattern isn’t immediately evident. The core pattern of Sea Pearls (2011) (center) consists of red and white circles of various sizes and color intensities that extend to the whole hyperbolic plane under symmetries of the original tessellation (7,7,7,7). The asymmetry of this pattern has the overall effect of breaking the original symmetry of the tessellation, which consists of four heptagons at each vertex. Sea of Pearls (2018) (top left) is a perfect example of how the broken symmetry of one tessellation, Sea Pearls (2011) can enhance the symmetries of another (3, 3, 3, 3, 3, 3, 3)."},{title:"Algorithmic Math Art",imgName:"algMathArt.png",altText:"",storyMedia1:"algMathArt1.png",storyMedia2:"algMathArt2.png",storyMedia3:"",storyMedia4:"",storyText:"The geometry and combinatorics behind tessellations provide the framework for creative, artistic expression. Sometimes, the choice of a particular motif can alter or hide the mathematical structure, obscuring or enhancing internal symmetries. Disoriented (top left), Disoriented and Confused (center), and Lost (bottom left) are based off hyperbolic tessellation (3, 4, 3, 4, 4) combined with patterns of increasing complexity and size. The mathematics and the algorithm behind the software Tess are determining the order in which the circles are drawn, hence the appearance of the artwork as a whole."},{title:"Sugaku-geijyutsu",imgName:"sugaku.png",altText:"",storyMedia1:"sugaku1.jpg",storyMedia2:"sugaku2.png",storyMedia3:"sugaku3.png",storyMedia4:"sugaku4.jpg",storyText:"Sangaku (top left) are geometrical theorems or problems painted on wooden tablets. Placed as offerings at Shinto shrines and Buddhist temples during Japan’s Edo period (1603-1868) by members of all social classes, Sangaku were often presented as mathematical solutions to questions or challenges to congregants. Sugaku-geiyutsu I (2018) (top left two) uses sangaku from Tashiro Shrine of Gifu Prefecture as a tile for (4, 4, 4, 4, 4, 4). The next tessellation, Sugaku-geiyutsu II (bottom left) uses a photo taken of the torii gates at Fushimi Inari, the head shrine of the god Inari in Kyoto, Japan using a pattern of seven triangles. Seven Towers (2011) (center) features a tessellation of the hyperbolic plane inspired by Japanese pagodas. Shown in a classical black, red, and white color scheme, the tessellation emphasizes a local, seven-fold symmetry."},{title:"Uniqueness and Existence",imgName:"nonUniqueF.png",altText:"",storyMedia1:"nonUniqueF1.png",storyMedia2:"nonUniqueF2.png",storyMedia3:"nonUniqueF3.png",storyMedia4:"nonUniqueF4.png",storyText:"The software Tess takes in a sequence of numbers and a vertex symbol and tries to construct a tessellation. The numbers in a vertex symbol -- say (3, 6, 4, 6) -- mean that around each point (vertex) of a tiling, there will be a triangle, a hexagon, a square, and a hexagon, in that precise order going around the point in the clockwise direction. The relation between the vertex symbols, and tessellations is intricate. Sometimes the tessellation corresponding to a symbol is unique and sometimes not. As the Tess code (bottom left) suggests, there are four different realizations of (3, 3, 3, 3, 3, 3, 5). Can you find the differences? Although there are infinitely many tessellations of the hyperbolic plane, not all vertex configurations form a tessellation e.g. (3, 5, 4, 5)."},{title:"The Twittering Machine",imgName:"hyperKlee.png",altText:"",storyMedia1:"hyperKlee1.png",storyMedia2:"hyperKlee2.png",storyMedia3:"hyperKlee3.png",storyMedia4:"",storyText:"Beautiful ideas and objects in mathematics can present an aesthetically appealing visualization. The Twittering Machine (2002) is inspired by the work of Paul Klee and predates Twitter. It’s a uniform tessellation (4, 4, 4, 6) in disguise with the structure of (4, 4, 4, 6) enriched with different tiles of various degrees of symmetry. The appeal of this kind of tessellation can lie in the sheer appreciation of its visual characteristics, a comprehension of the mathematics behind them, or the interplay of the two."},{title:"A Day in the Life of a Mathematician",imgName:"mathDay.png",altText:"",storyMedia1:"mathDay1.jpg",storyMedia2:"mathDay2.jpg",storyMedia3:"mathDay3.jpg",storyMedia4:"",storyText:"Mathematicians still use whiteboards to teach math? Yes, although blackboards and chalk are preferred for research and teaching. This hyperbolic and kaleidoscopic tessellation consists of six hexagons at each vertex, using the whiteboard in Dr. Radmila Sazdanović’s office to fill in the tiles. The edges of the tiles in the tessellation act as mirrors to reflect the board image."},{title:"Same or Different?",imgName:"sameDiff.png",altText:"",storyMedia1:"sameDiff1.png",storyMedia2:"sameDiff2.png",storyMedia3:"sameDiff3.png",storyMedia4:"",storyText:"These tessellations defined by the vertex symbol (3, 6, 4, 6) are the same; they merely appear to be different. This difference comes from choosing a different polygon as the focal point at the center of the disk: triangle, square, hexagon. Even stranger things happen in the last image on the left when the tessellation is not centered, but placed randomly inside the disk. These images provide the viewer with multiple perspectives of the same tessellation."},{title:"Networks and Squares",imgName:"networksSquares.png",altText:"",storyMedia1:"networksSquares1.png",storyMedia2:"networksSquares2.png",storyMedia3:"networksSquares3.png",storyMedia4:"networksSquares4.png",storyText:"The visualization of large information or social networks presents difficulties. Let's start with four friends at each corner of a square. Each of these four friends has four other friends, and these friendships overlap. One of your friend's friends is also friends with a different friend creating a confusing social network to be encoded into a grid. If each friend has five friends instead of four, the grid shape won’t work, as we would need five connections. And we would need even more room for visualizing a social network that is branched six ways. As the number of friends increases, a hyperbolic plane becomes the most optimal choice for visualizing these social networks because one can fit 5, 6, 7 or more squares around each point (see the figures). You can have as many friends as you wish."},{title:"Color Symmetry",imgName:"colSymm.png",altText:"",storyMedia1:"colSymm1.png",storyMedia2:"colSymm2.png",storyMedia3:"colSymm3.png",storyMedia4:"colSymm4.png",storyText:"Color is a method of tracking the symmetries of a tessellation. Many artists, such as Josef Albers, have noticed the essential interaction between colors and form. Different formal compositions of color guided by symmetries of a tessellation have a different visual impact and present different points of interest to a viewer. Which of these images presents the same information?"},{title:"The Harmony of the World",imgName:"harmony.png",altText:"",storyMedia1:"harmony1.png",storyMedia2:"harmony2.png",storyMedia3:"",storyMedia4:"",storyText:'Until Uranus was discovered in 1781, there were five known planets other than Earth. Coincidentally there are exactly five regular tessellations of the sphere, usually drawn as regular polyhedra. Johannes Kepler, the renowned 17th-century scientist, wrote an entire book (Harmonices Mundi, 1619) arguing that this was a physical law and connecting it to the theory of harmonics. By contrast, there are infinitely many "regular polyhedra" in a hyperbolic world. Tessellation (7, 7, 7, 7), consisting of four heptagons around each vertex, was used as a starting point for the top two tessellations, while the last one is based on octagons (8, 8, 8).'},{title:"Lava",imgName:"lava.png",altText:"",storyMedia1:"lava1.png",storyMedia2:"lava2.png",storyMedia3:"",storyMedia4:"",storyText:"The simple, bold designs of Lava (2012) (center), Volcano (2018) (top left), Eruption (2018) (bottom left) takes natural phenomena to the hyperbolic plane. Columnar jointing appears when lava flows into water, forming columns of polygonal cross-section in the tessellation. Similar ideas have also inspired technological breakthroughs in engineering and biomedicine such as negative stiffness honeycombs."},{title:"Knotted DiscoTess",imgName:"knotTess.png",altText:"",storyMedia1:"",storyMedia2:"",storyMedia3:"",storyMedia4:"",storyText:"Knotted DNA, proteins, light, shoelaces -- knots are important in our everyday lives and in science. With knotted tessellations of the hyperbolic plane, Knotted DiscoTess (2018) combines a research interest in knots with visualizations of the hyperbolic space. The knot used as a tile was created using KnotPlot by R. Scharain, then imported into the tessellation (5, 5, 5, 5, 5, 5, 5) consisting of seven pentagons at each vertex created using software by M. Christersson."},{title:"Something Old, Something New",imgName:"somethingOld.png",altText:"",storyMedia1:"somethingOld1.png",storyMedia2:"somethingOld2.png",storyMedia3:"",storyMedia4:"",storyText:"The importance and universality of visualizations is best seen through the lens of ethnomathematics; the same mathematical ideas appear across cultures all over the world. Ornamental art, basketry, and textiles -- one of the oldest implicit emanations of mathematical concepts -- are used as inspirations for the series of tessellations Something Old, Something New."},{title:"Eridanus",imgName:"eridanus.png",altText:"",storyMedia1:"eridanus1.png",storyMedia2:"eridanus2.png",storyMedia3:"eridanus3.png",storyMedia4:"",storyText:"Eridanus is the sixth-largest constellation, discovered by the Greek mathematician Ptolemy in the second century. Eridanus represents a river and these images pay an homage to Ptolemy and the beauty of the stars. This series of images, Eridanus (2012-2018) was created using the software Tess, tessellation (3, 4, 8, 4) made out of triangles, squares, and octagons, with patterns that exceed the size of the tile, forming an overlapping, flowing effect reminiscent of a stream."},{title:"Dum Spiro Spero",imgName:"spiro.png",altText:"",storyMedia1:"",storyMedia2:"",storyMedia3:"",storyMedia4:"",storyText:"Through its striking and radiant spiral, Dum Spiro Spero (2018) tells a story of hope. Constructed using an analogy of a spirograph in the hyperbolic plane, it depicts the rotation of a triangle around the red dot in the negatively curved hyperbolic plane. Hyperbolic geometry makes the pattern curvy and interesting. The pattern grows, unfolds, and opens up into an invitation to bridge the gap between words and pictures, and between math and art."},{title:"The Roads",imgName:"geodesic.png",altText:"",storyMedia1:"geodesic1.png",storyMedia2:"geodesic2.png",storyMedia3:"",storyMedia4:"",storyText:"The Roads (2018) is a triptych visualizing all the shortest roads between points in the hyperbolic plane represented in three different models: a Poincare or Klein disk, and a Poincare halfplane model. Models work as lenses. Lines representing roads look really different. Sometimes they’re straight and sometimes parts of circles much like an airplane’s polar routes, on which a flight from Philadelphia to Munich flies over the North Pole. No matter how they look, they’re always the shortest path (geodesic) even when they lead to infinity."},{title:"Fireworks",imgName:"3Models.png",altText:"",storyMedia1:"3Models1.png",storyMedia2:"3Models2.png",storyMedia3:"3Models3.png",storyMedia4:"",storyText:"In Fireworks series (2018) the same tessellation is featured in different models of the hyperbolic plane. Notice how very different the tessellation looks when viewed in disk or half-plane models of the hyperbolic plane. The main difference between the models is that infinity is represented either by the circle or the horizontal line at the bottom of the image. That’s why things appear small and fractal-like. 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e=this._targetAnimations();-1===e.indexOf(this)&&e.push(this)},_unmarkTarget:function(){var e=this._targetAnimations(),t=e.indexOf(this);-1!==t&&e.splice(t,1)}}}(n,r),function(e,t,n){function r(e){var t=c;c=[],e<m.currentTime&&(e=m.currentTime),m._animations.sort(i),m._animations=u(e,!0,m._animations)[0],t.forEach(function(t){t[1](e)}),a(),l=void 0}function i(e,t){return e._sequenceNumber-t._sequenceNumber}function o(){this._animations=[],this.currentTime=window.performance&&performance.now?performance.now():0}function a(){d.forEach(function(e){e()}),d.length=0}function u(e,n,r){g=!0,h=!1,t.timeline.currentTime=e,p=!1;var i=[],o=[],a=[],u=[];return r.forEach(function(t){t._tick(e,n),t._inEffect?(o.push(t._effect),t._markTarget()):(i.push(t._effect),t._unmarkTarget()),t._needsTick&&(p=!0);var r=t._inEffect||t._needsTick;t._inTimeline=r,r?a.push(t):u.push(t)}),d.push.apply(d,i),d.push.apply(d,o),p&&requestAnimationFrame(function(){}),g=!1,[a,u]}var 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