# Deep Spaces: Geometry Labs Bring Beautiful Math to the Masses [Slide Show]

They don't wear white coats or protective goggles, but mathematicians around the country have captured the imagination of students in ever more creative ways

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A model of the hyperbolic plane crocheted by students at the EAGL.....[
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A model of the hyperbolic plane crocheted by students at the EAGL.

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Sean Lawton/EAGL
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Middle school students learn about spherical geometry by drawing on balloons during a presentation by members of the Experimental Algebra and Geometry Lab (EAGL).....[
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When can a knotted-up loop of string be unknotted without cutting it? It turns out that this question is best studied by looking at the geometry of the space around the string rather than at the loop itself.....[
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When can a knotted-up loop of string be unknotted without cutting it? It turns out that this question is best studied by looking at the geometry of the space around the string rather than at the loop itself. This image is used to understand the hyperbolic geometry of this knot complement.

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Nathan Dunfield/SnapPy/Illinois Geometry Lab
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Both the octagon on the left and the L-shaped table on the right can be "folded" into genus-2 surfaces, which look like the thin layer of glaze on top of a two-holed donut. A team at the IGL is studying the trajectories of billiards on these surfaces.....[
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Both the octagon on the left and the L-shaped table on the right can be "folded" into genus-2 surfaces, which look like the thin layer of glaze on top of a two-holed donut. A team at the IGL is studying the trajectories of billiards on these surfaces.

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Jayadev Athreya/IGL
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An early 20th-century thread model from the archives of the math building at the University of Illinois at Urbana–Champaign. A group at the Illinois Geometry Lab (IGL) is documenting the models so they can be easily accessed and understood.....[
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An early 20th-century thread model from the archives of the math building at the University of Illinois at Urbana–Champaign. A group at the Illinois Geometry Lab (IGL) is documenting the models so they can be easily accessed and understood.

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Wendy G. Harris/IGL
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The Apollonian gasket is a fractal that arises when studying circle packing, which concerns how to most efficiently store round objects. This is an overlay of two different images that show different circle packings and how they interact.....[
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The Apollonian gasket is a fractal that arises when studying circle packing, which concerns how to most efficiently store round objects. This is an overlay of two different images that show different circle packings and how they interact.

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Tevis Tsai/EGL
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Hyperbolic geometry is one alternative to Euclidean geometry, which is taught in high school, and the hyperbolic plane is the analogue of the standard two-dimensional plane. In hyperbolic geometry, lengths are defined differently and lead to different looking "straight lines," which form the boundaries of polygons.....[
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Hyperbolic geometry is one alternative to Euclidean geometry, which is taught in high school, and the hyperbolic plane is the analogue of the standard two-dimensional plane. In hyperbolic geometry, lengths are defined differently and lead to different looking "straight lines," which form the boundaries of polygons. Here, an octagon in the hyperbolic plane has been stretched using a particular type of function called a loxodromic Möbius map, and the resulting image has been used to tile a region of a plane.

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Ryan Hoban/EGL
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A visualization created at the Experimental Geometry Lab (EGL) of how the complex plane changes when a certain function, called a Möbius transformation, is applied repeatedly. Möbius transformations preserve circles and angles, hence the face structure, but can distort size.....[
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A visualization created at the Experimental Geometry Lab (EGL) of how the complex plane changes when a certain function, called a Möbius transformation, is applied repeatedly. Möbius transformations preserve circles and angles, hence the face structure, but can distort size.

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Greg Laun/EGL
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A loxodrome is a spiral that arises from following a particular direction from the north or south pole of a sphere, meeting every line of latitude in the same (non-right) angle. This is a stereographic projection of a loxodrome onto a plane.....[
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A loxodrome is a spiral that arises from following a particular direction from the north or south pole of a sphere, meeting every line of latitude in the same (non-right) angle. This is a stereographic projection of a loxodrome onto a plane. A stereographic projection takes a sphere onto a plane in a prescribed way.

For more images like this one, visit Paul Nylander's website.

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Paul Nylander/Experimenal Geometry Lab (EGL)
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