Answers to the second set of 4D geometry practice problems:
1.Two ways to get the answer:
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Reason it out by thinking about the bases and connecting edges:
Each base has 6 vertices, so with two bases there are 12 vertices.
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If you're a really good artist, you can also draw it out and count (but bring colored pencils)
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2: First note that the 3D shape has 6 vertices and 12 edges
There are 6*2=12 vertices, and 12*2+6=30 edges
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Here's a picture to help me explain my answers:
(notice that I drew the two bases exactly the same in one color, and then connected the vertices using another color to help me keep track of what I was doing) |
There are 12 vertices. All of the vertices are either on the "top" base or the "bottom base". There are 6 vertices on each base, and two identical bases, so there are 6*2 vertices total There are 30 edges. There are 12 edges on each base, and there are two bases, so that makes 2*12=24 edges. Then there are edges connecting the vertices of the "top" base to the vertices of the "bottom" base--one edge (straw) for each of the "bottom" vertices. That makes 6 more edges: 24+6=30 edges. |
3. A. Explain why the number of vertices in the 4D prism is twice the number of vertices on the base
A 4D prism consists of 2 identical 3D bases, with extra edges connecting the corresponding vertices of the bases. All of the vertices are on the bases, and the bases are identical, so there are 2 x the number of vertices in one base.
B. Explain why the number of edges in the 4D prism is twice the number of edges on the base plus the number of vertices on the base.
To make a 4D prism, first you make two identical 3D bases, so if you know how many edges there are in one base, then there have to be twice that many edges in two bases. Then you connect corresponding vertices of the bases. That adds one more edge for each vertex on a base, so you that's why you add the number of vertices to twice the number of edges.
4. Alice is figuring out the number of vertices and edges in the 4D prism with base:
A. Alice knows that the 4D prism will have more vertices than the base, which has 5 vertices. How many more vertices will there be on a 4D prism? Why?
A 4D prism will have 5+_5___ vertices because... there are 5 vertices on the "top" base, and 5 more on the "bottom" base.
B. Alice knows that the 4D prism will have more edges than there are on the base, which has 8 edges. How many more edges will there be on a 4D prism? Why?
A 4D prism will have 8+_8__+_5__ vertices because... there are 8 edges on the "bottom" base, 8 edges on the "top" base, and 5 edges that connect the 5 vertices on the "bottom" base to the 5 vertices on the "top" base.
