MATHSEARCH 2000
Senior Project Prepared by Geoff Ball and Humphrey Gastineau-Hills, University of Sydney.

Pedan Polygons and Polyhedra - page 4

Section C. Research Exercises. 40 Marks.

Pedan Octahedral Antiprisms.
A different type of polyhedron can be formed from the same starting point as described in the Peterson-Jordan procedure outlined above. However, instead of the lateral faces being parallelograms, isosceles triangles are used. The orientation of each adjacent pair of lateral faces is reversed. That is, their bases are alternately on the top and bottom faces of the antiprism.
This can be envisaged by considering the case in which the image of an equilateral base triangle P₁P₂P₃, after rotation though 180° about its centre, and then elevated vertically, is the triangle P₁'P₂'P₃'. By joining the vertices as follows: P₁ P₃' P₂ P₁' P₃ P₂' P₁, one obtains 6 lateral faces which are isosceles triangles in which the orientation of the triangles is reversed in adjacent faces. Note that we have three pairs of parallel, oppositely oriented, lateral faces, and two horizontal faces (the triangles P₁P₂P₃ and P₁'P₂'P₃'

Exercise 8 (10 marks).
(i) Sketch the hexahedronal antiprism described above.
(ii) If the side length of the base equilateral triangle is 3, and the altitude of the antiprism is , determine the lengths of all sides and diagonals of the antiprism.

In the final exercises you are asked to explore some geometry of antiprisms.

Exercise 9 (20 marks).
(i) Show that each diagonal cross-section of a rectangular prism is a rectangle.
(ii) Determine the shape of the diagonal cross-sections e.g. P₁P₂P₁'P₂', of the Pedan octahedral antiprism described above.
(iii) Show that the vertices of the same Pedan octahedral antiprism lie on a sphere, naming its centre and radius.

In this next part, we will set up a three dimensional rectangular coordinate system with:
the centre of the sphere as the origin;
an axis, say the z axis, passing through the centroids of the horizontal triangles;
the y-z plane containing the altitudes of the horizontal triangles through P₁ and P₁'.

(iv) Determine the coordinates of the vertices of the antiprism in Exercise 8.

Exercise 10. (10 marks).
Consider a Pedan octagonal antiprism in which the horizontal triangles are right-angled triangles with the length of its hypotenuse c and other two sides a and b, satisfying the condition .

By introducing a rectangular coordinate system analogous to that in Exercise 9 (iv), determine the coordinates of the vertices of the antiprism in terms of a and c.

We hope you enjoyed this project and that it has helped you extend your mathematical horizons somewhat.

Please include a short note indicating if either of these outcomes were achieved.

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