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Example 4: Symmetries of a TetrahedronDescribe the symmetries of a tetrahedron_ A tetrahedron consists of four equilateral triangles that meet at six edges and four...

Question

Example 4: Symmetries of a TetrahedronDescribe the symmetries of a tetrahedron_ A tetrahedron consists of four equilateral triangles that meet at six edges and four corners (see Figure 9.4). Besides the motion leaving all corners fixed call it the 0" motion we can revolve 1209 or 2400 about a corner and the center of the opposite face (see Figure 9.4a), or we can revolve 180? about the middle of opposite edges (see Figure 9.4b). Since there are four pairs of a corner and opposite face and t

Example 4: Symmetries of a Tetrahedron Describe the symmetries of a tetrahedron_ A tetrahedron consists of four equilateral triangles that meet at six edges and four corners (see Figure 9.4). Besides the motion leaving all corners fixed call it the 0" motion we can revolve 1209 or 2400 about a corner and the center of the opposite face (see Figure 9.4a), or we can revolve 180? about the middle of opposite edges (see Figure 9.4b). Since there are four pairs of a corner and opposite face and three pairs of opposite edges; we have a total of 1 (0? motion) +4x2+3 = 12 symmetries. It is left as an exercise to check that these 12 symmetries are distinct and that no other symmetries exist. Figure 9.4 Symmetric revolutions of tetrahedron



Answers

Central Angle of a Tetrahedron A tetrahedron is a solid with
four triangular faces, four vertices, and six edges, as shown
in the figure. In a regular tetrahedron the edges are all of the
same length. Consider the tetrahedron with vertices
$A(1,0,0), B(0,1,0), C(0,0,1),$ and $D(1,1,1)$

$$
\begin{array}{l}{\text { (a) Show that the tetrahedron is regular. }} \\ {\text { (b) The center of the tetrahedron is the point } E\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right) \text { (the }} \\ {\text { "average" of the vertices). Find the angle between the }} \\ {\text { vectors that join the center to any two of the vertices (for }} \\ {\text { instance, } \angle A E B \text { ). This angle is called the central angle }} \\ {\text { of the tetrahedron. }}\end{array}
$$ [Note: In a molecule of methane $\left(\mathrm{CH}_{4}\right)$ the four hydrogen
atoms form the vertices of a regular tetrahedron with the carbon atom at the center. In this case chemists refer to the central angle as the bond angle. In the figure, the tetrahedron in the exercise is shown, with the vertices labeled $H$ for hydrogen and the center labeled $C$ for carbon. $]$

If we want to show that a tetra huge in, which is a solid with four triangular faces for overseas and six edges, is rectangular, meaning all the edges of the same length. Then we should first start out with our verte. Sees a being 100 be being 010 see being 001 and finally d being 111 So if you want to find the distance between two points, you can use the distance formula X two minus x one plus y tu minus Y one So we can say that D A distance is equal to root of one minus one squared, plus one minus zero squared, plus one minus zero squared to be equal to root. Two. Likewise, you can do the same with D B D. C A. C, maybe NBC, and see that all of these are swear to in the same manner. It's always one minus zero where zero minus one or zero minus zero. The numbers change, but these air all the exact same, just in different manners. From that, we can see that the lengths of all sides are the same. The given Tetra John is regular is what we're proving by doing this. So now that we know that it just regular we can ah find the angle between the vectors that join the center e of any two courtesies and weaken. Figure that out by saying that the vector E B is equal to zero minus 1/2 Which is he? Hi plus one minus 1/2 J plus zero minus 1/2. Okay, we can then simplify this out to E. B. Being equal to a negative 1/2 I positive 1/2 j and minus 1/2 K We can then do likewise for E A and say that he a is equal to positive 1/2 high, minus 1/2 J minus 1/2. Okay, we can see here that the I N J coordinates are effectively flip for A and B and then we can see that the angle between vectors E B and E A would be fatal. So we now know that CO sign of data is equal to you dot v over absolute value of you that absolute value of V and we can do the dot product of vector e V and E A to get negative you dot the is equal to negative 14 minus 14 minus 14 You can see that when you just be multiplying negative, half positive, 1/2 negative, 1/2 positive half and then two negatives and the two negatives on the back end. Be adding, you didn't be taking the route of E B and he a and with the root of be any a the 92 because they're effectively the same can say that it's the square root of native on half squared, plus 1/2 squared minus. They have 1/2 square. That comes out to 3 1/4 which is equal to root. 3/2. Andrew 3/2 for both of those stand. When we put that into this equation, we then know that co sign of data is equal to negative 1/4 either Route 3/2 times, Route 3/2. We can then get that out to be negative 1/3 and say that data is equal to inverse co sign of negative 1/3 which we could then no to be 109.47 degrees

Hello there in this problem as to give the number of axis of symmetry for a regular Paula he'd Ron. So we've got here is a tetra. He'd Ron basically a pyramid, and we want the number of axis of symmetry. And so we're gonna have, ah, a couple different types of access. Assume to remember, an axis of symmetry is just an access that we can draw through this figure. This three dimensional figure we can rotate it about that access and what we're gonna end up with is in 1 360 degree rotation, we're gonna end up with something that, ah, looks like what we started with. So they the image looks like the pre image. So with the Tetra Hydro on, we've got a couple of different kinds. We'll start with one that is probably the most obvious. If we draw on access, it goes through the Vertex and through the opposite face. So that's going to go through the face here and through overtax. And that's gonna be an access of symmetry. And so we're gonna have one of those call that a vertex Teoh face. Now, if we think about how many of those were gonna have or have each projects that the opposite face. There's one that's drawn. Then we'll end up with one here at this Vertex, one of this vortex one of this for Texas and each one of those you and go to the opposite face. And so we'll end up with four Those total one drawn plus 2 +34 So there's four total that our vertex to face clean it up a little bit. No, Put that back. So for those and now we're gonna have another type of access here, and that is the, um, midpoint of the opposite sides. This one is a little bit harder to picture, but oh, it's gonna be We have an opposite edge here and opposite edge here. We draw this line through that, and then we spend that along. That line will end up with, uh, you know, rotational symmetry on that line two's that's a, um over texting. Each one of those are access. I'm sorry of rotational symmetry, so that would be an axis of rotational symmetry. And how many of those were gonna have? It's gonna be opposite midpoint to the opposite. Midpoint will have won their love one that goes through the opposites here and then one that goes to the opposites here, right through the backs. There's three of those. Tell me. Get rid of this will say midpoint to midpoint about mid to mid. Save me some writing mid to mid and what we saw. We had one. We have to and then we have three of those. Let's write that down three. Now the total number of axes of symmetry is just gonna be the sum of a way happier. So remember he started out with the Vertex to face and there were four of those. And then we have the midpoint of the side to the opposite end point of the side. And there three of those. So there's some there is seven, and that will be our answer. There are seven, and he's

What we have to do. In part a of this problem is find the volume of the Tetrahedron a regular tetra radio. Sorry. So we're given the four points from Problem 50 in section 9.4 and we use them to formulate three vectors so we can calculate the scaler triple product and then multiply that result by 1/6 to get a volume. All I've done here is Set D is the terminal point in all three cases and A, B and C as the initial points. To calculate these three vectors, what we're gonna do to start office, find the cross product beady cross CD. So if we do that, we get Bt two times. CD three is equal to zero time zero, which is just zero minus b t three times CD two, which is just one times one. So that's one comma Bt three times. CD one is just one times one so one minus bt one time see three is just one time zero, which is zero and the final part baby. One time CD too, is just one times one. So that gives one minus. B two times CD one zero times one is zero. So now for our cross product, we get an answer of negative one comma, one comma. What? And from here, we can find the doc product 80 dot be across CDs. So when we do that, we get a d dot Let's just write this vector 80 dot negative one comma, one comma one, we get negative. One times zero is zero plus one times one is just one and one more time. One times one is simply one. And this sum is equivalent to What we've found right now is the scaler triple product is equal to two. And to find the volume of the Tetra hater And all we have to do is multiply this by 1/6 Get V equals to six or 1/3 inches. Cute. So this is there a final answer?


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