MTH2132 The Nature and Beauty of Mathematics: Homework problem set 3, 2023
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MTH2132 The Nature and Beauty of Mathematics: Homework problem set 3, 2023
1. Vital statistics a) How many vertices, edges, triangular faces and square faces does a snub cube have?
To find the answer to this question, you should use the fact that the snub cube is constructed from the cube by exploding and twisting its faces and filling in the resulting gaps with equilateral triangles.
And you should calculate the numbers we are after in terms of the numbers of the ver- tices, edges, and faces of a cube. Only writ- ing down numbers that you looked up some- where will get you no marks :)
b) Do the same for the following semi-regular polyhedron. You can think of this semi- regular polyhedron as a modified dodecahe- dron, with its twelve regular 10-gon faces es- sentially distributed around the polyhedron as the regular pentagons in a dodecahedron.
c) I think it is important that all of you own copies of the five regular solids, and therefore one of the assignments this week is for all of you to get/make a set, take a photo of yourself or your student ID with this set, and submit this photo as proof that you are really the proud owner of all five.
There are many ways in which you can make a set. Check out the following links for some ideas:
nets
follow the links to the nets with tabs, e.g. http://tinyurl.com/l86yd8f
the dodecahedron calendar is one of these
nets put to good use
amazing world maps (these are really, re- ally beautiful :)
straws work really well for the tetrahedron, octahedron, and icosahedron
dice
origami
Rubik’s cubes
balloons
2. 4-d truncations Now let’s try to do the same for one of those 4d monsters. The 120- cell has 120 dodecahedral cells, 720 pentago- nal faces, 1200 edges, and 600 vertices. Also, there are four edges ending in each vertex.
If we partially truncate the 120-cell, we ar- rive at a semi-regular 4d polytope with two types of 3d polyhedra as cells. One is a regu- lar polyhedron, the other is one of the semi- regular polyhedra.
a) What are those two types of cells?
b) How many cells, faces, edges, and vertices does this partial truncation have? Again, you should calculate the numbers we are af- ter in terms of the numbers of cells, faces, edges and vertices of the 120-cells. Again, only writing down numbers that you looked up somewhere will get you no marks :)
Hint: You can use the 4d Euler formula vertices − edges + faces − cells = 0.
c) What are the numbers for the full trun- cation of the 120-cell?
3. Putting Euler’s formula to work. Draw 9 (pink) vertices and connect them as shown in the following diagram.
a) Prove that the resulting graph/network consisting of 9 vertices and 15 edges is not planar, that is, it cannot be drawn such that no two of the 15 edges intersect away from the vertices.
Use the same line of argument that we used to prove that the water-electricity-gas graph is not planar by applying Euler’s formula.
Hint: If this graph was planar it would con- tain at most one triangular face (a face hav- ing 3 vertices connected by three edges). Why?
b) Which of the following networks are pla- nar? Explain!
4. Networks on doughnuts. As we know neither the gas-water-electricity graph nor the complete graph on 5 vertices is planar. This implies that neither can be drawn on the surface of a sphere without some of the edges also intersecting away from the ver- tices.
The following diagram illustrates that it is possible to draw the complete graph on 5 vertices on the surface of a torus (= dough- nut) without some of the edges intersecting away from the vertices.
Here the torus is being represented by an orange ring. (Parts of) edges drawn solid lie on the part of the torus facing you and those drawn dashed on the part of the torus facing away from you.
a) Can the graph in 3 a) be drawn like this on the surface of a torus without some of the edges intersecting away from the vertices.
b) Which of the graphs in 3 b) can be drawn like this on the surface of a torus without some of the edges intersecting away from the vertices.
If it is possible to draw these graphs on the surface of a doughnut without some of the edges intersecting away from the vertices, draw pictures with the torus being repre- sented by a ring. If it is not possible, give a proof of this fact.
2023-04-21