Compound of Five Octahedra
This polyhedron is created to look like 5 octahedra that pass through each other. In reality the center of this structure is hollow. In the photo above you can see 5 of the yellow octahedron's 6 vertices.
Begin by using the template below and cutting one net out of each color. You will need 5 colors, one for each of the apparent octahedra. In total you will be cutting 6 nets of each of the 5 colors.
Fold each piece along the fold lines and glue the two edges that are on the bottom of the picture above together to form a 4-sided pyramid. The base of the pyramid won't be part of this pyramid. But, if the base were, it would be a rhombus. The pyramid will have two planes of symmetry. Let's say that the major axis of symmetry is the longest axis.
Take 5 of these small pyramids, one of each color, and glue them into a ring having their longest axis of symmetry radiating outward as you can see in the photo below.
These first 5 pyramids will help you realize where the next pieces should be placed.
Your finished polyhedron will consist of 2 different arrangements of pyramids. The first arrangement is the one demonstrated in the photo above with 5 pyramids surrounding a central point. The second arrangement is of 3 pyramids glued with their minor axis radiating from a central point. Here's what they look like.
Continue surrounding that initial 5 pyramid construction with 3 pyramid constructions. It should then be clear which colors are necessary to complete the subsequent 5 pyramid arrangements.
Notice that the yellow and orange pyramids from the first view are also in this second arrangement of pyramids. The third pyramid to be glued to this arrangement is the same color as the pyramid that is opposite the valley that connects the orange and yellow pyramids, blue.