Bing's house 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:

It has 4 quadrapod vertices connected by 8 tripod curves (two loops and 6 edges).

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

I am interested in surfaces with the same type singularities.

(Motivated by this question.)