I made another model of 30 pencils with icosahedral symmetry. This model is based on 10 triangular prisms. It can be considered as the dual of the previous model posted previously (based on 6 pentagonal prisms). If you rotate each pencil by a certain angle and allow them to pass through each other, you go from one model to the other.
I made a model of 12 interlocking regular dodecahedra. It has the icosahedral rotational symmetry (without reflection symmetry). I use 6 colors, 2 dodecahedra per color. It was built with straws and paper clips.
The aperiodic Tile(a, b) can be constructed as a 4D skew nonplanar polygon then projected to 2D. Changing the projection direction, we can get the whole continuum including hat, turtle, and the non-curvy spectre. For some directions, we get a self crossing polygon in 2D.
In the tiling mixing hats and turtles (Fig 3.1 of "A chiral aperiodic monotile"), all 4D tiles are the same skew polygon but in different orientations. At the same time, some tiles are projected to hats, some turtles.