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**Actualizing Mathematics:**

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

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**Present and Future Projects**

Tiling Penrose Tile Project Blacksmithing Domed Ceiling Szalassi Polyhedron Landscaping Osculating Circles Elliptical Stairs Desargue’s Theorems Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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Penrose Tiles Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**UM-Dearborn Alumni Wall Mosaic**

Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Special Figures, Empires**

Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Penrose Tiles in Public Places**

Bachelor Hall, Miami University 22 ft in diameter, tiles 10.5 x 17 inches Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Penrose Tiles in Public Places**

Carleton College 15 ft in diameter Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Penrose Tiles in Private Places**

Alex Feldman’s shower floor Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Penrose Tiles at UM-Dearborn**

7 ft square Custom tiles from… Pewabic Pottery of Detroit Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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Copper Foyer Dome Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**What shape to cut the “wedges” from 2x4 sheets of copper?**

Kyle first made a profile, then a concrete form that he would pound the sheet copper into. He knew he would need nine segments, or wedges to complete the dome, but did not know how to cut them before hand so that, when bent into form, they would mate “simply”--read that as “linear”, “clean”. Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem**

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**Szilassi Polyhedron 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 faces vertices edges holes Tetrahedron 4 6 Szilassi Polyhedron 7 14 21 1 Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Public Szilassi Commercially available thru Hans Schepker Lighting**

A UM-Dearborn courtyard sculpture? Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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Face information Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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Edges with mass Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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Wall & Patio Designs A number of years back we put an addition on our home. It required us to rework our retaining walls. We opted for concentric circles, stepped back, echoing the kitchen bow window. We already had a pattern established in the old portion of the patio and needed something to complement it in the new. My first thought was prompted by the phase plane portrait of a pendulum. Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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Wall & Patio Execution Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Pendulum Phase Plane Portrait**

Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Osculating Circles, Decaying Sines**

Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Osculating & Concentric Circles**

Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Osculating & Concentric Circles**

Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Elliptical Stair Existing Wouldn’t this be nice?**

Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Elliptical Stair Existing Proposed**

Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Desargue’s Theorem Let three lines lie on a common point.**

Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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Desargue’s 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. Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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Desargue’s 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 A’B’, on the lines AC and A’C’, and on the lines BC and B’C’ … Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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Desargue’s 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 A’B’, on the lines AC and A’C’, and on the lines BC and B’C’ lie on a common line. Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Desargue’s Dual 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 A’B’, on the lines AC and A’C’, and on the lines BC and B’C’ 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 A’B’, thru the points AC and A’C’, and thru the points BC and B’C’ lie on a common point. Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Desargue’s Dual Theorem**

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

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**Desargue’s Dual 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. Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Desargue’s Dual 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 A’B’, thru the points AC and A’C’, and thru the points BC and B’C’ … Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Desargue’s Dual 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 A’B’, thru the points AC and A’C’, and thru the points BC and B’C’ lie on a common point. Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Desargue’s Theorem, Dwarf Orchards**

Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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**Desargue’s Theorem, Dwarf Orchards**

Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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“Doing” mathematics Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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