Making blocks from four cubes
Supplies:
Lots of wooden cubic inch blocks, wood glue.
Task:
Design and build all the possible blocks consisting of four cubes.
What constitutes a block of four cubes?
Each cube must share at least one face with another cube.
What does it mean to say that two blocks are
different? We say that two solids are the same if one could replace the
other.
How many blocks are there? How many cubes do you need for a whole set?
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Answer to the question, how many blocks are there?
Showing that there are 8 such blocks can be done in different ways.
Examples.
(1) There are only two 3-cube blocks.
We can adjoin one cube to the first pattern in 7 different ways, forming 4 one-level blocks (square, L, Z, T), and 3 two-level blocks (corner, left-hand man, right-hand man).
There is only one block that can be obtained from the second 3-cube block that cannot be obtained from the first.
Note that the right-hand man and the left-hand man are not the
same, because one cannot replace the other. But they are mirror images of
each other. In geometry, people use the concept of congruent futures.
Two mirror images are considered to be congruent.
(2) Draw graphs where the vertices represent cubes and the edges indicate that cubes have a face in common. No such graph contains a triangle because 3 cubes cannot touch each other.
There are 3 graphs possible.
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