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The angles meeting at a point must sum to 360 degrees so as to leave no gaps nor result in overlaps.
For a regular shape, therefore, each vertex must be a factor of 360 degrees. The only such values are 60 deg (triangle), 90 deg (square) and 120 deg (hexagon). There is more scope with irregular shapes.
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RossBecause overlaps and gaps within the plane aren't allowed, by definition. Think of the simplest example of a non-tessellating shape, a circle. How could you fill up a 2-D plane with no overlaps or gaps with just circles? Even using fractal analysis you can't since both the Apollonian gasket and the 5 circles inversion fractal have Hausdorff dimensions less than 2, which would be the number needed to fill a 2-D plane.
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How do you create a quilt only using geometric shapes?
Select a shape that tessellates. Some shapes will tessellate by themselves, others will tessellate in pairs (octagons and squares), or larger groups. See the link for a flavour.
How 3D shape tessellate?
Some 3D shapes will tessellate as for example a brick wall
Why can only some shapes tessellate?
Shapes can tessellate only if a number of them can meet at a point and cover 360 degrees without overlap. For regular shapes this requires that the angles of the shape are a factor of 360 degrees. For non-regular shapes it is necessary that the angles of the shapes can be grouped so that they sum to 360 degrees.
Why do some shapes tessellate and some don't?
Tessellation is repeating a pattern over and over and filling an area with no overlaps and no gaps. Some shapes can't do this, because they don't fit with themselves without leaving big gaps between the parts.
Can every hexagon tessellate?
No, only some can.
