As it turns out, there are only three regular polygons that can be used to tessellate the plane: regular triangles, regular quadrilaterals, and regular hexagons. Is a regular hexagon made up of 6 equilateral triangles? A hexagon is made up of 6 congruent equilateral triangles. Each equilateral triangle has a length of 8 units.
Can a regular polygon tessellate a plane?
A regular polygon can only tessellate the plane when its interior angle (in degrees) divides 360 (this is because an integral number of them must meet at a vertex). This condition is met for equilateral triangles, squares, and regular hexagons. Furthermore, which figure Cannot Tessellate?
How many tessellations can a regular triangle have?
There are only three shapes that can form such regular tessellations: the equilateral triangle, square, and regular hexagon. Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps.
What types of planes can be tesselated?
A plane can be uniformly tesselated by regular triangles, squares, and hexagons. A hyperbolic plane can be uniformly tesselated by heptagons or apeirogons, each regular polygon meeting two others at a vertex. There are many other regular polygonal tilings. [ 2]
What is tessellation in geometry?
Tessellation refers to the arrangement of shapes closely fitted together, especially of polygons in a repeated pattern without gaps or overlapping just as in tiling. Equilateral triangles, squares and regular hexagons are the only regular polygons that will tessellate.
Can an equilateral triangle tessellate a plane?
Some shapes can be used to tessellate the plane, while other shapes cannot. For example, a square or an equilateral triangle can tessellate the plane (in fact any triangle or parallelogram can), but if you try to cover the plane with a regular pentagon, you'll find there's no way to do it without leaving gaps.
Which polygon will tessellate a plane?
Only three regular polygons (shapes with all sides and angles equal) can form a tessellation by themselves—triangles, squares, and hexagons.
Can regular octagons and equilateral triangles tessellate the plane?
Equilateral triangles, squares and regular hexagons are the only regular polygons that will tessellate.
How do you know if a regular polygon can tessellate the plane?
A regular polygon can only tessellate the plane when its interior angle (in degrees) divides 360 (this is because an integral number of them must meet at a vertex). This condition is met for equilateral triangles, squares, and regular hexagons.
What makes tessellation possible for equilateral triangles?
A shape will tessellate if its vertices can have a sum of 360˚ . In an equilateral triangle, each vertex is 60˚ . Thus, 6 triangles can come together at every point because 6×60˚=360˚ . This also explains why squares and hexagons tessellate, but other polygons like pentagons won't.
What are the only regular polygons that tessellate?
There are only three shapes that can form such regular tessellations: the equilateral triangle, square and the regular hexagon. Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps.
Which polygon will not tessellate a plane?
pentagonsIn fact, there are pentagons which do not tessellate the plane. The house pentagon has two right angles. Because those two angles sum to 180° they can fit along a line, and the other three angles sum to 360° (= 540° - 180°) and fit around a vertex.
Will regular octagons tessellate?
Does a regular octagon tessellate? First, recall that there are in a pentagon. Each angle in a regular pentagon is 1080 ∘ ÷ 8 = 135 ∘ . From this, we know that a regular octagon will not tessellate by itself because does not go evenly into .
Does a hexagon tessellate with a triangle?
An equilateral triangle is 60° therefore 60° x 6 = 360° hence six equilateral triangle would tessellate. A hexagon is 120° therefore 120° x 3 = 360° hence three hexagons would tessellate. Any shapes with 4 sides (quadrilaterals) can tessellate as where the vertices meet it must equal 360°.
What kinds of triangles work in a tessellation?
Equilateral triangles have three sides the same length and three angles the same. Can you make them fit together to cover the paper without any gaps between them? This is called 'tessellating'.
How many sides can a polygon have to form a regular tessellation?
Since the interior angles get larger as the number of sides in a polygon gets larger, no regular polygons with more than six sides can tessellate by themselves.
What shapes make regular tessellations?
There are three regular shapes that make up regular tessellations: the equilateral triangle, the square and the regular hexagon.
Explorations
Some Basic Tessellations
- Recall that a polygon is a closed plane figure made by joining line segments.You might want to review the relevant material in Fundamental Conceptsconcerning polygonsbefore reading this section. The fundamental question we will discuss in this section is: More precisely, which polygons can be used as the only tile in a monohedral tessellation of the plane? Before moving o…
Tessellations by Quadrilaterals
- Recall that a quadrilateral is a polygon with four sides. To prove, divide a quadrilateral into two triangles as shown: Since the angle sum of any triangle is 180°, and there are two triangles, the angle sum of the quadrilateral is 180° + 180° = 360°.Taking a little more care with the argument, we have: <math>\alpha_1 + \delta_1 + \gamma = 180^\circ</math> and <math>\alpha_2 + \delt…
Tessellations by Convex Polygons
- Every shape of triangle can be used to tessellate the plane. Every shape of quadrilateral can be used to tessellate the plane. In both cases, the angle sum of the shape plays a key role. Since triangles have angle sum 180° and quadrilaterals have angle sum 360°, copies of one tile can fill out the 360° surrounding a vertex of the tessellation. The ...
Tessellations by Regular Polygons
- Recall that a regular polygon is a polygon whose sides are all the same length and whose angles all have the same measure.A regular <math>n</math>-gon has <math>n</math> equal angles that sum to <math>(n-2)180^\circ</math>,so: The table shows the corner angles for the first few regular polygons:
Relevant Examples from Escher's Work
- Fundamental forms of regular division of the plane, Visions of Symmetrypg. 33
- Sketch #A7 (Regular division with triangles)
- Tessellation by triangles, sketch (2) from the abstract motif notebook, Visions of Symmetrypg. 83.
- Sketch #131-134 (Pentagon tessellations), and Tiled Column, New Lyceum, Baarn
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