Every face is an identical regular polygon

 

 

 

 

Math 4250/6250: The dot product, the point groups, and the regular solids.
Definition 1 A polyhedron P ⊂ R
3
is a regular solid if every face is an identical regular polygon
and the same number of faces meet at each vertex.
You probably remember that there are only 5 regular solids: the tetrahedron, cube, octahedron,
dodecahedron, and icosahedron, and I suspect that you can picture them. But here’s a harder
question: what are the coordinates of their vertices?
One way to generate coordinates is to use the point groups. For instance, any four (noncoplanar) points ~p, ~q, ~r, ~s ∈ R
3
form a tetrahedron by taking the four triangular faces to be {~q, ~r, ~s},
{~p, ~r, ~s}, {~p, ~q, ~s}, and {~p, ~q, ~r}, as below.
~p
~q
~r ~s
Since three triangles meet at each vertex, the second condition of Definition 1 is met regardless of
the positions of ~p, ~q, ~r, ~s. However, the triangular faces may not all be equilateral.
~p
~q
~r
~s
~e1 ~e2
~e3
Here, ~p = (1, 1, 1), ~q = (−1, −1, 1), ~r = (1, −1, −1) and ~s = (−1, 1, −1). This is a special
tetrahedron!
1
1. (10 points) We are now going to use the point group G to show that the tetrahedron above with
~p = (1, 1, 1), ~q = (−1, −1, 1), ~r = (1, −1, −1) and ~s = (−1, 1, −1) is a regular solid.
(1) (5 points) Prove that A and B permute ~p, ~q, ~r, and ~s and write down the permutation.
Conclude that every element of G permutes these vectors in some way (because it is a
product of A’s and B’s).
2
3
(2) (5 points) Use the last question to find an isometry in G (that is, a product of A’s and B’s)
which takes the edge {~p ~q} to each of the other 5 edges of the tetrahedron: {~p ~r}, {~p ~s},
{~q ~r}, {~r ~s}, and {~s ~r} to show that all the faces are equilateral triangles.
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2. (10 points) Starting with any ~v = (1, x, 0) (assume x < 1), we can generate 12 vectors
~v1, . . . , ~v12 by applying the 12 matrices in G to ~v. We can group these into the vertices of
3 rectangles in the ~e1 − ~e2, ~e2 − ~e3 and ~e3 − ~e1 planes as below.
~e1
~e2
~e3
I~v = (1, x, 0)
A~v = (0, 1, x)
Label each vertex above with its coordinates and the corresponding matrix in G (written as
a product of A’s and B’s). We have labeled I~v and A~v above to help you get started.
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3. (15 points) As we did with the tetrahedron, we’re now going to use the point group to show
that certain distances between our 12 points are the same and we’re going to connect this group
to a different Platonic solid!
(1) (5 points) The edge {~v A~v} is one of 12 edges marked in blue on the picture below. Use
the results of Question 2 to describe each of these edges in the form {C~v D~v} where C
and D are products of A’s and B’s.
I~v
A~v
There is plenty of space to write computations below and on the next page, but it might
be easier for you to write in the coordinates on the picture above.
6
7
(2) (5 points) Prove that all of these edges have the same length by finding isometries in
G which take {I~v A~v} to each of these edges. This proves that the blue triangles are
equilateral. Hint: You’ll eventually need to use the relations between products of A and
B that you developed from (AB)
3 = I in the last homework.
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(3) (5 points) You don’t have to match isometries with edges explicitly again, but the 12
isometries in G map the edge {~v, BAB~v} to the 12 edges in red below left. The red
edges all have the same length and the red triangles are equilateral. So our construction
yields a one-parameter family of solids which are G-symmetric, depending on the x in
~v = (1, x, 0).
Each has 12 vertices and 20 triangular faces, with 5 triangles meeting at every vertex.
However, while the 4 red triangles and the 4 blue triangles are always equilateral, the 12
green triangles are only isoceles. An example is shown below center.
Solve for the length of the red edges r(x) and the blue edges b(x) to prove that b(x) =
r(x). Then set b(x) = r(x) = 2x (the length of the short side of the rectangles) to find
the x which makes the green triangles equilateral and the entire figure an icosahedron, as
shown below right.
Hint: The value x should be familiar . . . what is it?

Sample Solution

processing model also referred to as the Dual Process Model of Coping with Bereavement by Stroebe and Schut (Death Studies, 1999), is a natural process that helps us to find a balance between facing the reality of the loss (loss-orientation) and learning to re-engage with life after the loss (restoration-orientation). It is in finding the balance may explain why grief is often described as an emotional roller coaster. Many people experience a back and forth between both loss-orientation and restoration-orientation responses, for example moving between classic grief reactions, crying, anger, depression etc. and learning how to manage finances, form new relationships and taking on roles that the dead person may have done, for example looking after the children. It is in the restoration-orientation phase that grieving people may focus on day-to-day tasks and get temporary relief from the emotional drain of the loss. It is possible to get a sense of the dual processing model when working with C as she is angry and depressed but has days where she is able to focus on the new baby’s imminent arrival. With further work once the baby has arrived, as so not to put any unnecessary stress or upset upon C before the baby is born, it may be possible to encourage her to explore her own dual process of grief, alongside learning how to care for her baby with the support of the staff at the mother and baby unit. N has experienced a normal dual process of grieving as she has days where she is feeling low especially when it comes to significant events such as birthdays, anniversaries and holidays, but has expressed that she is more positive towards Christmas this year as she has her daughter and the new baby living at home with her, therefore she is able to focus on them rather than her losses. It is my intention to work alongside N to encourage her to explore her own dual process. Therese Rando (1993) developed the six R’s of grieving and according to Rando, in order to achieve the six R’s, a person must Recognise the loss by acknowledging and understanding the death, React to the separation by experiencing the pain, give some form of expression to the psychological reaction of the loss and identify the secondary losses, Recollect the relationship with the loved one by remembering them realistically and re-experience the feelings they had for them, Relinquish the old attachments to the deceased, Re-adjust and adapt to their new world without forgetting their old one by developing a new relationship with the deceased, adopt new ways of being in the world without their loved one and form a new identity and finally Re-invest. Rando stated that complicated mourning is present whenever there is some compromise, distortion, or failure of one or more of the six “R” processes of mourning. It is clear from working with C that her grief is complicated and so some sort of compromise in one or more of the six R’s is probable

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