**Geometrical Gems (2021)**

Prepared by:

Joseph Malkevitch

Department of Mathematics

York College (CUNY)

Jamaica, New York 11451

email:

__malkevitch@york.cuny.edu__

web page:

__http://york.cuny.edu/~malk/__

The list below consists of a collection of geometrical theorems that I feel all K-12 students should see at some point during their public school education. The results are known to various degrees and I have included a very brief statement or explanation of why I find these particular examples special. They are listed in no particular order.

1. Euler's traversability theorem

Starting at any vertex, it is possible to tour the edges of a graph once and only once returning to the start vertex if and only if the graph is connected and has an even-number of edges at each vertex (even-valence).

2. Euler's polyhedral formula

If G is a plane connected graph or P is a 3-dimensional convex polyhedron than the numbers of vertices, faces, and edges of G or P satisfies vertices + faces - edges = 2.

3. Klee/Chvatal/Fisk Art Gallery Theorem

If P is a plane non-self-intersecting (simple) polygon with n verticess then floor (n/3) guards (sensors) placed at vertices are sometimes necessary and always sufficient for the guards to see all the points of P and its boundary.

4. Bolyai-Gerwien-Wallace Theorem

Given two plane simple polygons of the same area it is possible to cut each of them into a finite number of polygonal pieces which can be reassembled (in the style of a jigsaw puzzle) to the other.

5. Sylvester-Gallai Theorem

Given a finite set of points in the plane not all on the same line there is some pair of these points that determines a line with no other point of the set on this line.

6. Steinitz's Theorem

The set of graphs which arise as the vertex-edge graphs of a convex 3-dimensional polyhedron are the planar 3-connected graphs.

Comment: This result allows one to study the combinatoirla properties of 3-dimensional convex polyhedra witout visiualizing or making models of them in 3-space but by looking at a special class of geometrical objects in the plane.

7. Morley's Theorem

For any triangle in the plane the trisectors of its angles intersect in points which are vertices of an equilateral triangle.

8. Jordan Curve Theorem

A simple closed curve in the plane parttions the plane into three sets of points: those on the curve, those inside the curve and those outside the curve.

9. Taxicab geometry obeys all of the Hilbert Axioms for Euclidean geometry except for the "congruence" axiom.

10. Curves of constant breadth h all have the same perimeter

11. Partition types of convex quadrilaterals and 3-dimensional tetrahedra.

12. Mani's Theorem

Let G be a planar 3-connected graph (which by 6. above we know can be represented in 3-dimensional space as a bounded 3-dimenstional strickly convex polyhedron) with automorphism group H (its collection of combinatorial symmetries) H. Conclusion: G can be represented by a convex polyhedron whose isometry group (distance preserving symmetries) is isomorphic to H.