### Engaging math activities for the summer break - Day 16

**What's the task?** The task is to combine several 3- and 4-pyramids into larger 3- and 4-pyramids.

**What's the setup?** You'll need 4 tetrahedra and 6 square pyramids. Having 8 tetrahedra and 8 square pyramids will allow to complete 3- and 4-pyramids simultaneously.

For your convenience, here are the maps of the pyramids. Just cut, fold, and glue to obtain the pyramids.

And smaller variants:

**An observation to make** Once you succeed in making larger 3- and 4-pyramids, imagine continuing the process *recursively*. The whole space can be filled with such 3- and 4-pyramids. This is possible because the two shapes have all edges equal.

**An important auxiliary point** When a linear measurement of a plane figure changes by a factor of 2, the area changes by a factor of 4. For a 3D figure, a change by a factor of 2 in a linear measurement leads to a change in volume by a factor of 8.

**Compare volumes** of 3- and 4-pyramids with all the edges equal. The previous note allows for computing the ratio of the volumes of 3- and 4-pyramids. It is so happens that the latter is exactly twice the former.

For the construction of 3-pyramid, see online interactive simulation, and another one for the construction of a 4-pyramid.

Two 4-pyramids joined base-to-base give a regular octahedron. It could be observed that the 3-pyramid is composed of 4 smaller tetrahedra and 1 octahedron (all with equal edges.) On the other hand, an octahedron consists of 6 smaller octahedrons and 8 tetrahedra - again, all with equal edges. This remark shows that regular tetrahedra and octahedra that have equal edges tessellate the whole of the 3d space.