More fun than a hypercube of monkeys

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This Model Is Featured In Figure 3.31 Of [Visualizing Mathematics With 3D Printing](Http://3Dprintmath.Com). This Sculpture Was Inspired By A Question Of [Vi Hart](Http://Www.Youtube.Com/User/Vihart). As Far As We Know, This Is The First Sculpture (In Fact, Physical Object) With The Quaternion Group As Its Symmetry Group. The Quaternion Group {1,I,J,K,-1,-I,-J,-K} Is Not A Subgroup Of The Symmetries Of 3D Space, But It Is Very Naturally A Subgroup Of The Symmetries Of 4D Space. The Monkey Was Designed In A 3D Cube, Viewed As One Of The Eight Cells Of A Hypercube. The Quaternion Group Moves The Monkey To The Other Seven Cells. Radial Projection Moves The Monkeys Onto The 3-Sphere, The Unit Sphere In 4D Space, Then Stereographic Projection Moves The Monkeys To 3D Space. The Distortion In The Sizes Of The Monkeys Comes Only From This Last Step - Otherwise All Eight Monkeys Are Identical. For More Details, See Vi'S And My [Paper](Http://Archive.Bridgesmathart.Org/2014/Bridges2014-143.Html), Or Evelyn Lamb'S [Blog Post](Http://Blogs.Scientificamerican.Com/Roots-Of-Unity/2014/05/19/A-Hypercube-Of-Monkeys-Quaternion-Group/) At Scientific American. This Is Joint Work With [Will Segerman](Http://Www.Willsegerman.Com). Also Available From [Shapeways](Http://Shpws.Me/T2Sb). Also Check Out The Interactive Animated Online Version At [Monkeys.Hypernom.Com](Http://Monkeys.Hypernom.Com). (Use The Wasd Keys, Arrow Keys, Numbers 1-6.)

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