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Actualizing Mathematics: Osculating Circles, Coved Ceiling, Desargues Duals, Penrose Tiles, and Szilassi Sculptures in Public and Private Gardens Michael Lachance Professor of Mathematics UM-Dearborn March 2009

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Tiling –Penrose Tile Project Blacksmithing –Domed Ceiling –Szalassi Polyhedron Landscaping –Osculating Circles –Elliptical Stairs –Desargues Theorems Present and Future Projects Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Penrose Tiles Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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UM-Dearborn Alumni Wall Mosaic Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Special Figures, Empires Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Penrose Tiles in Public Places Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem Bachelor Hall, Miami University 22 ft in diameter, tiles 10.5 x 17 inches

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Penrose Tiles in Public Places Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem Carleton College 15 ft in diameter

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Penrose Tiles in Private Places Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem Alex Feldmans shower floor

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Penrose Tiles at UM-Dearborn Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem 7 ft square Custom tiles from… Pewabic Pottery of Detroit

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Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem Copper Foyer Dome

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What shape to cut the wedges from 2x4 sheets of copper? Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Name coined by Martin Gardiner, November 1978 Each of its faces touches all the other faces Euler's formula: f+v-e=2-2h Seven-color conjecture http://www.drking.plus.com/hexagons/szilassi/index.html Szilassi Polyhedron facesverticesedgesholes Tetrahedron4460 Szilassi Polyhedron 714211 Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Commercially available thru Hans Schepker Lighting A UM-Dearborn courtyard sculpture? Public Szilassi Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Face information Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Edges with mass Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Wall & Patio Designs Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Wall & Patio Execution Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Pendulum Phase Plane Portrait Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Osculating Circles, Decaying Sines Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Osculating & Concentric Circles Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Osculating & Concentric Circles Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Elliptical Stair Wouldnt this be nice?Existing Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Elliptical Stair ProposedExisting Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Desargues Theorem Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem Let three lines lie on a common point.

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Desargues Theorem Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem Let three lines lie on a common point. Draw points A and A on one line, points B and B on a second line, and points C and C on a third line.

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Desargues Theorem Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem Let three lines lie on a common point. Draw points A and A on one line, points B and B on a second line, and points C and C on a third line. The points on the lines AB and AB, on the lines AC and AC, and on the lines BC and BC …

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Desargues Theorem Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem Let three lines lie on a common point. Draw points A and A on one line, points B and B on a second line, and points C and C on a third line. The points on the lines AB and AB, on the lines AC and AC, and on the lines BC and BC lie on a common line.

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Desargues Dual Theorem Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem Let three lines lie on a common point. Draw points A and A on one line, points B and B on a second line, and points C and C on a third line. The points on the lines AB and AB, on the lines AC and AC, and on the lines BC and BC lie on a common line. Let three points lie on a common line. Draw lines A and A thru one point, lines B and B thru a second point, and lines C and C thru a third point. The lines thru the points AB and AB, thru the points AC and AC, and thru the points BC and BC lie on a common point.

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Desargues Dual Theorem Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem Let three points lie on a common line.

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Desargues Dual Theorem Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem Let three points lie on a common line. Draw lines A and A thru one point, lines B and B thru a second point, and lines C and C thru a third point.

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Desargues Dual Theorem Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem Let three points lie on a common line. Draw lines A and A thru one point, lines B and B thru a second point, and lines C and C thru a third point. The lines thru the points AB and AB, thru the points AC and AC, and thru the points BC and BC …

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Desargues Dual Theorem Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem Let three points lie on a common line. Draw lines A and A thru one point, lines B and B thru a second point, and lines C and C thru a third point. The lines thru the points AB and AB, thru the points AC and AC, and thru the points BC and BC lie on a common point.

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Desargues Theorem, Dwarf Orchards Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem http://www.mtholyoke.edu/courses/jmorrow/Desargues%2 7Theorem.htmlhttp://www.mtholyoke.edu/courses/jmorrow/Desargues%2 7Theorem.html

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Desargues Theorem, Dwarf Orchards Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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Doing mathematics Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

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