1. 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

2. 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

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

4. UM-Dearborn Alumni Wall Mosaic
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

5. Special Figures, Empires
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

6. 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

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

8. Penrose Tiles in Private Places
Alex Feldman’s shower floor
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

9. 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

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

11. 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

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

13. 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

14. 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

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

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

17. 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

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

19. Pendulum Phase Plane Portrait
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

20. Osculating Circles, Decaying Sines
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

21. Osculating & Concentric Circles
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

22. Osculating & Concentric Circles
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

23. Elliptical Stair Existing Wouldn’t this be nice?
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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

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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. Desargue’s Theorem, Dwarf Orchards
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

35. Desargue’s Theorem, Dwarf Orchards
Domed Ceiling Szalassi Polyhedron Osculating Circle Elliptical Stair Penrose Tile Desargue’s Theorem

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