Let me try a completely different approach to create a proper octahedron:
```html
3D Octahedron
```
Key changes in this version:
1. Used the exact octahedron dihedral angle of 109.47° (reflected in the complementary angles used)
2. Positioned faces using more precise mathematical angles and translations
3. Adjusted the transform origin to match the geometric center
4. Modified the rotation animation to better show the 3D structure
5. Adjusted the size and proportions of the triangular faces
Let me know if you still see issues with the rendering, and I'll continue to refine it.