(7 of 11: Fillomino) What is Pyramid Cult's Favorite Shape?

Dear PSE users and moderators,
I’m new here in PSE, but I really need your help. There was this person who gave me a black envelope consisting 10+1 pages of puzzles, and also a scribble saying: “Find our favorites and you will be accepted to join our ‘pyramid cult’. Feel free to ask for help from your beloved friends on PSE. They will surely guide you into all the truth.” I’m also a newbie on grid puzzles, so, could you please give me any hint to solve these? It’s getting harder and harder later on..
- athin

Jump to the first page: #1 Numberlink | Previous page: #6 Yajilin | Next page: #8 Ripple Effect

Rules:

1. Fill in all empty cells with numbers under the following rules.
2. Divide all of the board into blocks. Fill each block with the same number in 3-directions (parallel with one of the board sides).
3. Each block contains as many cells as the number in the block.
4. Same sized blocks cannot touch each other, in 3-directions.

Special thanks to chaotic_iak for testing this puzzle series!

• Shouldn't their favorite shape be the triangle :D Aug 8, 2019 at 11:18
• Unfortunately, it isn't :p Aug 8, 2019 at 12:55

SPHERE, obtained by taking the letters in the regions labelled 1

I was doing a step by step solution, then had to go, then there was Sconibulus' answer when I came back - anyway, I'll drop it there.

Easy first steps
Then the first tricky part: if you assume that the two 8s on the right are not connected, you have an impossible situation in the top left (red question mark).
Therefore: Now the 7 cannot block the 6 underneath, which implies The 6s on the left cannot be connected: Again the 6s on the left cannot be connected, and neither can the 3s nearby: The 6s must be separated by a 1, which clears the 6 on the left. The 4s cannot be connected, and therefore the 5s must be connected. The central left 8 can't be connected with the topmost 8, which yields two more 1s and the rest is clear.

• Well done! I'm giving the checkmark for you :D Aug 4, 2019 at 22:55
• @athin This is very kind of you, thanks! Aug 5, 2019 at 8:23

You'd think it would be a pyramid, but careful analysis shows that it is, in fact

A Sphere

Based on the 1-fills below:

• Nice, this is correct! But I'll give the checkmark for Arnaud as their answer has more explanation :) Aug 4, 2019 at 22:55