the **polychoron** (actually a 10-choron or 10-cell) of which you can see the **development** or **net** in 3D, has the following cells:

**2C + 2H + 2A + 4B**

**C** is a cube of edge 2

**A** is an offcenter pyramid type A with square base & lateral faces (all right trigs):

2 x (2,2,2sqrt(2)) + 2 x (2,2sqrt(2),2sqrt(3))

**B** is an offcenter pyramid type B with sq. base & lat. faces (all right trigs):

2 x (2,2sqrt(2),2sqrt(3)) + 2 x (2,2sqrt(3),4)

it is well known that 3 off-center pyramids type A **glued around an edge of length 2sqrt(3)** make a **cube** (C)

I use the symbol **H** (heptahedron) to mean the non-convex solid made of 2 glued A-pyramids with a 2*sqrt(3) common edge, so

**2C + 2H + 2A + 4B** = 2 cubes, 2 heptahedra, 2 A-pyramids, 4 B-pyramids (pay attention 1H + 1A = 1C)

now, the **decachoron** spoken of above is a non-convex **half-tesseract** (or a half 4-hypercube) & 2 equal such hypersolids glued appropriately in 4D make a **tesseract** (or **8-cell** or **4D-hypercube**)

pay attention that a gluing in 4D between 2 identical cells make those disappear inside the 4th dimension (just as when you in 3D have 2 ordinary square pyramids & glue them through their bases, the result is a solid with 8 trig. faces,i.e. the 2 bases were glued out inside the solid); in the present case the 2 hypersolids are such that they can be glued simultaneously through 4 B-pyramids so that there disappear 8 B-pyramids at the same time, inside the final tesseract.

in short

2C +2H +2A +4B +

2C +2A +2H +4B = 2C +2C =4C; 2H+ 2A =2C; 2A+ 2H =2C & 4B+ 4B= nil (glued out inside the 4D-hypercube)

[it is a bit long, so please read it again if needed]