# How many times can I cut a shape in half before getting a new shape?

If you take triangle and slice it from one corner to the opposite base, you get another triangle. That leaves you with two new triangles. If I cut those triangles in half, I'll have four.

On the other hand, if I cut a circle in half, I immediately have a different shape.

Is there a formula/ name of concept that would tell me how many times I could cut a shape in half before I no longer receive that shape?

Also, for example, would the formula take cutting an equilateral shape in half and getting another equilateral triangle (which I don't think is possible)? Or does it only take into account how many times you can cut the equilateral shape in half and get another triangle (e.g. acute or right)?

• If you can do it once, you can do it infinitely many times. One example would be the triangle formed by cutting a square along a diagonal. – Jens Mar 23 '19 at 15:54
• You would need to rigously define "shape" and "cut in half" first, otherwise the question can not be answered. – Somos Mar 23 '19 at 17:36
• @Somos I can see how to define "cut in half" in more detail (e.g. cut in half from one corner or edge to another vs. just cut the area in half like when you cut a square in half using diamond). But what do you mean by defining "shape" more rigorously? – JustBlossom Mar 23 '19 at 17:44
• I mean exactly what I wrote. what do you mean by "shape"? Does it mean exactly the same set or what? Are two circles in different places and sizes the same "shape"? Are two triangles in different places and rotated the same "shape"? Do you know Banch-Tarski paradox? – Somos Mar 23 '19 at 17:45
• What Somos is trying to get at is what do you mean by saying that two shapes are the same? Yes, you can cut a triangle as described, but the new triangles are not the same size, nor are they even similar to the old. So by saying you "get the same shape", you have chosen to ignore size and dismilarity, and focus only on "closed curve consisting of 3 joined line segments" as being the definition of "shape". Cut a circle in two, and you get two closed convex curves, just like the original. Maybe that could count as meaning of "same shape"? It is a matter of what you decide "same shape" means. – Paul Sinclair Mar 24 '19 at 1:57