These are two α-polyhedron discovered by Robin Houston: the 45° dodecahedron, and the 135° octahedron. These polyhedra have only right angles except for one (called α). They relate to a conjecture that states that the α angle can only take algebraic values. The two polyhedra can combine to form a rectangular prism.

I wanted to print these after watching this video:

I found that the shapes are available on Sketchfab, but the obj files contained errors that prevented me from slicing and printing them. So I went back to the sources ( https://www.unhyperbolic.org/sydler.html, https://www.e-periodica.ch/digbib/view?pid=com-001:1965:40::7#71 ) and remodeled them from scratch. 

Print it as you like, those models are very forgiving.

Source files can be found at : https://cad.onshape.com/documents/8013828bdbd6565137623e11/w/ce481d58d4bb2a84a71d3a7b/e/1235fb4b1981b7fc8036ed4c?renderMode=0&uiState=666748dfcf423210aca46d70

Notice: Any attribution of this project should also mention Robin Houston.

PS: In the meantime the model on the Sketchfab page got fixed, so for those who want to download it from Robin Houston directly, please get it here : https://sketchfab.com/3d-models/dodecahedron-eb31e9f575b4486c9e949a9e7d8f72d1