Answer:OPTION 3
Children’s blocks lie scattered on the floor. You start playing with them — squares, rectangles, triangles and hexagons — moving them around, flipping them over, seeing how they fit together. You feel a primal satisfaction from arranging these shapes into a perfect pattern, an experience you’ve probably enjoyed many times. But of all the blocks designed to lie flat on a table or floor, have you ever seen any shaped like pentagons?
People have been studying how to fit shapes together to make toys, floors, walls and art — and to understand the mathematics behind such patterns — for thousands of years. But it was only this year that we finally settled the question of how five-sided polygons “tile the plane.” Why did pentagons pose such a big problem for so long?
Quantized Academy
Patrick Honner, a nationally recognized high school teacher from Brooklyn, New York, introduces basic concepts from the latest mathematical research.
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To understand the problem with pentagons, let’s start with one of the simplest and most elegant of geometric structures: the regular tilings of the plane. These are arrangements of regular polygons that cover flat space entirely and perfectly, with no overlap and no gaps. Here are the familiar triangular, square and hexagonal tilings. We find them in floors, walls and honeycombs, and we use them to pack, organize and build things more efficiently.
These are the easiest tilings of the plane. They are “monohedral,” in that they consist of only one type of polygonal tile; they are “edge-to-edge,” meaning that corners of the polygons always match up with other corners; and they are “regular,” because the one tile being used repeatedly is a regular polygon whose side lengths are all the same, as are its interior angles. Our examples above use equilateral triangles (regular triangles), squares (regular quadrilaterals) and regular hexagons.
Step-by-step explanation:
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