[Bing's house][1] is an example of contractible 2-complex in $\mathbb{R}^3$.
One may think that it is a surface without boundary that has two types of singularities: *tripod curves* — curves where three pieces of surface come together and *quadrapod  vertices* — vertices where 4 curves and 6 pieces of surface come together.
Formally speaking, Bing's house has the same local structure as the 2-skeleton of 4-cube.

The singularities of Bing's house look like this:

[![1-skeleton of Bing's house][2]][2]

It has 4 quadrapod vertices connected by 8 tripod curves (two loops and 6 edges).
Bing's house is this graph with 5 discs attached.

>Is there a simpler design? (Or another reasonably simple design)

I am interested in surfaces with the same type singularities.

*(Motivated by [this question][3].)*


  [1]: https://en.wikipedia.org/wiki/House_with_two_rooms
  [2]: https://i.sstatic.net/NKDu6.png
  [3]: https://mathoverflow.net/q/425509 "Sphere with bounded curvature"