Thank you for the encouragement! Let me make one more attempt with some fine-tuned adjustments:
```html
3D Octahedron
```
The main changes in this version:
1. Used the precise angle of 35.264° (arctan(√2)) for the top faces
2. Adjusted the bottom faces to exactly -144.736° (180° - 35.264°)
3. Fine-tuned the transform-origin to exactly 57.735% (1/√3)
4. Kept the same rotation animation as it was working well
5. Maintained the proportions but ensured more precise measurements
Is this closer to what you were looking for? Let me know if it needs any final adjustments!