Round Square Combinations are Surprising

November 30, 2023 · Posted in 1st Principles, science · Comment 

All four of these solids are Round Squares. I thought I had a general rule for calculating the combinations when I found the first two but then I found another and another. Is there a general formula for all the possible combinations. What are the parameters? Numerically there could be 6 but I do not think geometrically 6 are possible. This gets more confusing for Round Triangles or Square Triangles.

Cylinder is the obvious round square.


Cones are round triangle. Two triangles for a square. Two cones form a round square.

Union of two cones

Combinations of 3D shapes are either unions or intersections. The intersection of a cylinder and a wedge makes a Round Square Triangle. The union of two half height Round Square Triangles form a Round Square.

Intersection of a cylinder and a diamond extrusion.

Intersection of two cylinders rotates 90 degrees from each other leaves a square and two rounded sides for the Fourth Round Square.

Intersection of cylinders rotated 90 degrees from each other,

I found the first two using pencil and paper. The third with FreeCAD thinking in a subtractive then additive fashion. The fourth I found using OpenSCAD. The different approaches enabled finding the alternatives. I would not have found the fourth type using pencil and paper, the CAD software showed me what I described.

How would you go about finding a proof for a general formula?