# Geometry statement

by 0rthodontist
Tags: geometry, statement
 Sci Advisor P: 1,253 I have devised the following statement and a short proof of it: No polygon may be tiled by convex quadrilaterals, each of which shares exactly one face with the polygon. Is this known or trivial, or where could I find out if it is?
 P: 15,319 Not that I doubt you, but are there any further conditions that you are assuming, such as -no convex polygon -a single convex quadrilateral As it stands, your polygon could be any shape (such as a star), and you could use any or many of an infinite range of shapes and sizes of quadrilaterals to tile.
 Sci Advisor P: 1,253 The polygon does not have to be convex and since each quadrilateral shares a side with the polygon, there will be as many quadrilaterals as the polygon has sides. There is no restriction on the shapes or sizes of the quadrilaterals, other than that they must be convex.
 Emeritus Sci Advisor PF Gold P: 4,500 Geometry statement EDIT: Never mind, I missed the word "convex"
 Sci Advisor P: 1,253 One thing that is not clear is whether this is also true if you use convex n-gons for n > 4 instead of quadrilaterals. I would suspect that it is.
 P: 15,319 "...since each quadrilateral shares a side with the polygon, ..." My bad. I misread your post. I thought it said a MAXIMUM of one face (as in, you could have entirely internal quads that share zero faces). In fact it says EXACTLY one face. So, point retracted. Further clarification, though this may be bifurcating bunnies: I'm not sure the wording makes it clear that a quad shares the ENTIRE face of the polyogn, i.e. does it exclude two or more quads sharing the same polygonal face?