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Actualizing Mathematics: Osculating Circles, Coved Ceiling, Desargues Duals, Penrose Tiles, and Szilassi Sculptures in Public and Private Gardens Michael.

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Presentation on theme: "Actualizing Mathematics: Osculating Circles, Coved Ceiling, Desargues Duals, Penrose Tiles, and Szilassi Sculptures in Public and Private Gardens Michael."— Presentation transcript:

1 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

2 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

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

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

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

6 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

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

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

9 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

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

11 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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13 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 Szilassi Polyhedron facesverticesedgesholes Tetrahedron4460 Szilassi Polyhedron Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargues Theorem

14 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

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

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

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

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

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

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

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

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

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

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

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

26 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.

27 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 …

28 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.

29 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.

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

31 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.

32 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 …

33 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.

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

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

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


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