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

Pedan Polygons and Polyhedra - page 3

Exercise 5 (8 marks).

(i) Verify that the quadrilateral P₁P₂P₃P₄ is cyclic.
(ii) Using (i), find the length of P₁P₄
(iii) Show that the quadrilateral P₁P₄P₅P₆ is an isosceles trapezium.
(iv) Verify that the hexagon P₁P₂P₃P₄P₅P₆ is a Pedan hexagon.

Clearly, by deleting a vertex of a Pedan hexagon, and all sides and diagonals incident to it, one can construct a Pedan pentagon. However, as the following example illustrates, one can construct a Pedan pentagon directly.

Exercise 6 (16 marks).
(i) The lengths of the parallel sides P₁P₃ and P₅P₄ of the isosceles trapezium P₁P₃P₄P₅ are 7 and 4 respectively, with the slant sides have length 6. Verify that this trapezium is Pedan trapezium.
(ii) We-will construct a convex pentagon P₁P₂P₃P₄P₅ based on the trapezium in (i): The vertex P₂ is located by constructing P₅P₁P₂ P₁P₅P₆.
Prove that P₁P₂P₄P₅ is concyclic and in fact that P₁P₂P₄P₅ is an isosceles trapezium.
(iii) Prove that: (a) P₂P₃P₄ P₂P₁P₅;
(b) P₁P₂P₃P₄P₅ is a Pedan pentagon.

B2. Pedan Polyhedra.
Pedan pyramids are easily constructed from Pedan polygonal bases since one can make the slant height an integer and there are no diagonals to worry about apart from those already covered by the construction of the base.

There appears to be no single practical way of generating all Pedan polyhedra. Despite that, there are various procedures which generate some families of Pedan polyhedra.

Peterson and Jordan described the following process.

• Step 1. Replicate a Pedan rectangle by rotating a copy of it through 90° about its centre.
• Step 2. Raise the top copy vertically and form a convex polybedron whose other (lateral) faces are congruent parallelograms. Notice that the vertical elevation provides an infinite number of possible integer slant heights.
• Step 3. Select the dimensions cunningly so that the resulting solid is a Pedan hexahedron (six-faced figure).

Exercise 7 (14 marks).
Begin with a rectangle with side lengths a and b (a < b). Raise the copy (rotated through 90° about the centre of the rectangle) to a height H and join the relevant vertices to form a hexahedron.
(i) Show that if c is the slant height of the hexahedron then
(ii) Express the length of the internal diagonals of the hexahedron in terms of a, b and c.
(iii) Show that one can form a Pedan hexahedron based on an 8 by 15 rectangle.

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