Topic 1 Subject introduction: Discovery guide
You have been given two identical toy tetrahedrons with coloured edges: in each, three of the edges are a single colour and the other three edges vary in colour. Place each of your tetrahedrons so that the three monochromatic edges become the base, and orient your tetrahedrons so that the three edges of varying colours appear in the same places.
Our goal is to explore the symmetry of the tetrahedron by manipulating one of the tetrahedrons while leaving the other tetrahedron in a steady “reference” position. For the purpose of comparing the final orientation of the manipulated tetrahedron to the orientation of the stationary reference tetrahedron, it will be more useful for us to keep track of vertices rather than edges. To assist with this, think of the vertices of your tetrahedrons as being “labelled” with their “adjacent” colour (ignoring the repeated colour around the base). Assign the label “white” to the vertex at the peak where the varying colours meet.
For simplicity in this introduction, we will focus only on rotational symmetry and ignore reflectional symmetry for now. By rotational symmetry we mean the following.
Question 1.0.3.
In what ways can we rotate the second tetrahedron so that, after rotation, its geometric orientation in space is still identical to the stationary reference tetrahedron, even though its orientation of colours may have changed?
Warning 1.0.4.
In the activities below, always return your two tetrahedrons to being identically oriented both geometrically and with respect to the colour-coded vertices before considering a completely new rotation.
Discovery 1.1.
Consider an axis of rotation running through the peak vertex out through the centre of the base.
(a)
Considering only angles of rotation that are less that \(2 \pi\) in magnitude, there are five rotations around this axis that result in the two tetrahedrons remaining in identical geometric orientation. What are the angles of these five rotations?
(b)
Why not bother considering angles of rotation of magnitude greater than \(2 \pi\text{?}\) Why not bother considering angles of rotation of magnitude exactly \(2 \pi\text{?}\)
(c)
Considering only the differences in final orientation of the colour-coded vertices from reference tetrahedron to rotated tetrahedron, we actually only need to consider three rotations around the current axis, not five. Why?
(d)
If we begin to think algebraically about these rotations as well, we might consider that we really only need to think about one particular rotation around the current axis, as the other two rotations can be “built” from that one special rotation. What do you think this means?
Discovery 1.2.
In Discovery 1.1, we counted three rotations around the vertex-face axis through the peak vertex. There are four vertices, so there are four such vertex-face axes, which brings us up to twelve rotations total right?
Wrong! Figure out why the above paragraph is over-counting and determine the true number of vertex-face-axis rotations. Make sure you are only considering final orientation of colour-coded vertices; do not take axis or angle of rotation into account at all.
The over-counting problem is not related to the “building” of one rotation from another considered in Task d of Discovery 1.1.
The rotations we have considered so far correspond to certain aspects of the symmetry of the tetrahedron. If we performed one of these rotations with the colour-coding of the vertices removed, when the rotation was complete no one would be able to tell that we had manipulated the tetrahedron at all. The same would not be true if we applied one of these rotations to an irregularly-shaped object, so these rotations can be thought of as somehow encoding some particular aspect of the symmetry of the tetrahedron.
You may have noticed that the axis for each rotation considered so far also has symmetry-related properties: each of these axes passes through a vertex and the centroid of the opposite side. Thinking of a vertex as the “centre” of itself, we can consider each axis so far to pass through the “centre-points” of two oppositely-faced components of the tetrahedron.
The point of view considered in the previous paragraph will help you figure out another type of axis of rotational symmetry for the tetrahedron.
Discovery 1.3.
(a)
Determine another “centre-to-centre” axis through the tetrahedron besides the vertex-face axes we've already considered. Make sure it is possible to rotate the tetrahedron around that axis so that it returns to its original geometric orientation at some point before a full rotation of \(2 \pi\) radians; otherwise, discard that axis and try something else.
(b)
What angles of rotation about this axis return the tetrahedron to a geometrically-identical position?
(c)
How many different final colour orientations can we achieve through such rotations?
Discovery 1.4.
In total, there are three axes of rotation in the tetrahedron of the type you discovered in Discovery 1.3. Again, considering only final orientation of colour-coded vertices after a rotation, what is the total number of rotations around such axes in the tetrahedron?
Discovery 1.5.
(a)
Once again, considering only final orientation of colour-coded vertices after a rotation to ensure you are not over-counting, what is the total number of rotational symmetries of the tetrahedron you have discovered in Discovery 1.2 and Discovery 1.4?
(b)
Make a table describing all the rotations. Include the following information, described relative to the colour-coding of the vertices in the reference tetrahedron: axis of rotation; angle (use only non-negative angle values); vertices that remain fixed (if any); pairs of vertices that swap positions (if any)
Now let's continue some of the algebraic thinking we began in Task d of Discovery 1.1.
Discovery 1.6.
(a)
Choose a vertex-face axis of rotational symmetry. What happens if you perform the same rotation around that axis twice in a row, without returning the tetrahedron to its “reference” orientation in between? What if you repeat that rotation three times? Four times?
(b)
What algebraic pattern in the angles is occurring when you repeat a single rotation in this way?
(c)
Repeat for an edge-edge axis of rotational symmetry.
Discovery 1.7.
Now choose both a single vertex-face axis and a single edge-edge axis.
(a)
Perform a rotation around one of the axes and then the other, without returning the tetrahedron to its reference orientation in between.
IMPORTANT: The second axis does not rotate with the tetrahedron while you are rotating around the first axis. It may help to think of your chosen axes as being “frozen” in position in your reference tetrahedron.
Can you find a single rotation in your table from Discovery 1.5 that rotates your reference tetrahedron into the same final orientation of colours as your double-rotated tetrahedron?
(b)
Repeat Task a but with the order of the two rotations reversed. Is the result the same or different?
Discovery 1.8.
(a)
For each of the rotations that you tabulated in Discovery 1.5, can you find a rotation in your table that, if performed consecutively with the given rotation (similarly to Discovery 1.7), returns your tetrahedron back to the reference orientation?
(b)
Does order matter when performing the combined rotations in Task 1.8.a?
(c)
In each of the pairs you've identified in Task a, what existing mathematical terminology do you think you would give to the two rotations in that pair relative to each other?
Now let's consider two new shapes.
When considering their rotational symmetry, keep in mind that these are three-dimensional figures, and can be rotated in three dimensional space.
Discovery 1.9.
The hexagonal plate also has twelve rotational symmetries. Describe each of the eleven non-trivial ones in terms of axis and angle of rotation.
Discovery 1.10.
The dodecagonally-based pyramid also has twelve rotational symmetries. Describe each of the eleven non-trivial ones in terms of axis and angle of rotation.
Discovery 1.11.
Each of the three shapes we have considered today has exactly twelve rotational symmetries. Yet clearly the geometric nature of the symmetries that each shape possesses is different from that of the other two shapes. What are some of the ways that these differences are evident in the “algebra” of performing multiple rotations consecutively? What similarities are there in the patterns of “algebra” between pairs of shapes?