What is the term for a shape that has uniform dimensions?

Question

Is there a term to describe a shape that has uniform dimensions?

Squares and circles would fall into this category for 2D shapes, as they are as wide as they are tall. Rather than ellipses or rectangles, since these can be wider than tall and vice verse.

Similarly, cubes and spheres would have this attribute for 3D shapes.

This term would also be quantitative. Some shapes would have more or less of this attribute.

Symmetrical comes to mind, but that doesn't seem accurate. For example, the shape of a tree is symmetrical, but often much taller than wide.

Cheers

Answer 1

The phrase "uniform dimensions" is very vague, confusing, and open to numerous interpretations.

I think you're seeking Meissner bodies, of which Reuleaux triangles are the most famous example.

See also here.

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As a mini-tutorial on how to ask better questions, if I understand your question, here would be a better way to state it:

What is the mathematical term for a convex shape in $\mathbb{R}^n$ that spans the same maximum distance in one direction regardless of the orientation of that object?

Answer 2

Firstly what you describe is not at all something used in mathematics, that is an "intuitive" description that you want, not a mathematics thing.

I seem to have gotten two elements from your text:

• You don't want a shape wider than taller
• You want "uniform" shapes, but I am not sure what you mean by that

For the first one, one can say that it depends on the coordinate because you list "cubes" as one of those "uniform" shapes whatever that means. If you rotate the cube 30°, one side will be longer in a certain axis.

Therefore, what I understood, your criteria describes "shapes in N-dimension in an orthonormal basis that are invariant with rotation of 90° along the axis of the base".

Answer 3

As @PackSciences says, this (vaguely expressed) property does not have an generally accepted name.

Perhaps you are interested in figures for which the bounding rectangle with minimum area is a square (or hypercube). Perhaps a hypercube with edges parallel to coordinate axes.