Q: Can a regular heptagon tessellate a plane with no overlaps or gaps?

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yes no yes no

Triangle, square and hexagon

Equilateral triangle, square and regular hexagon.

No. Equilateral heptagons (7 sided figures) do not tessellate the plane. Not if no other polygons are allowed. But if you allow a (non-equilateral) pentagon then you might be able to tessellate the plane!

No not normally

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No.

yes no yes no

Triangle, square and hexagon

Equilateral triangle, square and regular hexagon.

Equilateral triangle, square and regular hexagon.

No. Equilateral heptagons (7 sided figures) do not tessellate the plane. Not if no other polygons are allowed. But if you allow a (non-equilateral) pentagon then you might be able to tessellate the plane!

No not normally

yes

A regular octagon can tessellate the plane when combined with regular squares. By placing a square in the center of the octagon and surrounding it with eight octagons, the shapes can be repeated infinitely, filling the plane without gaps or overlaps

The rectangle is the simplest and most obvious case of a geometrical form that can tile a plane.

No, it is not true that you cannot tessellate a six-sided polygon by itself. Hexagons are a type of polygon that can tessellate, which means they can be arranged in a repeating pattern to completely cover a plane without any gaps or overlaps.

No