Can a tiling be constructed with triangles and regular pentagons?

Can a tiling be constructed entirely with equilateral triangles?

yes

Can regular pentagons and regular hexagons tile a flat surface?

Regular pentagons cannot tile a flat surface without leaving gaps, as their internal angles (108 degrees) do not allow for a perfect fit. In contrast, regular hexagons can tile a flat surface efficiently because their internal angles (120 degrees) allow them to fit together perfectly without any gaps. Thus, while hexagons are capable of tiling, pentagons are not.

Why do triangles make a tiling?

Because they can tessellate

Can regular pentagons and regular hexagons fit together to rip a flat surface?

Yes, regular pentagons and regular hexagons can fit together to tile a flat surface. This combination can create a tessellation pattern where the pentagons and hexagons alternate, filling the space without any gaps. However, it requires careful arrangement and specific angles to achieve a seamless fit, as the internal angles of these shapes are different. Generally, this type of tiling is more complex than using just one type of polygon.

Can regular heptagons and equalateral triangle tile a flat surface?

Equilateral triangles can tile a plane, but regular heptagons cannot; nor can they tile the plan together. Where vertices meet (at a point on the plane) there is a complete turn of 360°. Each vertex of an equilateral triangle is 60°; 360° ÷ 60° = 6, a whole number of times, so a whole number of equilateral triangles can meet at a vertex of the tiling. Each vertex of a regular heptagon is 128 4/7°; 360° ÷ 128 4/7° = 2 4/5 which is not a whole number, so a whole number of regular heptagons cannot meet at a vertex of the tiling, so there will be gaps. With one regular heptagon there are 360° - 128 4/7° = 232 3/7°, but this cannot be divided by 60° a whole number of times, so one regular heptagon and some equilateral triangles cannot meet at a vertex of the tiling without gaps. With two regular heptagons there are 360° - 2 x 128 4/7° = 102 6/7°, but this cannot be divided by 60° a whole number of times, so two regular heptagons and some equilateral triangles cannot meet at a vertex of the tiling without gaps. With three or more regular heptagons, they will overlap when trying to place them on a plane around a point - leaving no space for any equilateral triangles.

Which three regular polygons can tessellate in a plane?

The three regular polygons that can tessellate in a plane are equilateral triangles, squares, and regular hexagons. These shapes can fill a space without any gaps or overlaps because their interior angles are divisors of 360 degrees. Equilateral triangles have angles of 60 degrees, squares have angles of 90 degrees, and regular hexagons have angles of 120 degrees, all of which allow for complete tiling of the plane.

Which regular polygons will fit together to tile a surface?

It depends on the shape of the surface Flat surface can be tiled by triangles, squares, and hexagons, these are the only combinations for the regular tessellations. semi-regular tessellations (where multiple polygons are used in the same tiling) There are in fact an infinite number of possible tessellations. All polygons can work from triangles to approaching a circle... a circle tiling would require an infinite number of infinitesimally small polygons around it, so you may or may not consider this a possibility. NOT all polygons can be in the same tessellations, for example triangles, heptagons, and 42-gons cannot be in a 1:1:1 ratio. In 3 dimensions regular polygons can be perfectly assembled into only 5 regular polyhedrons (3d version of polygon) (the platonic solids - these have been used to represent the elements, fire water, earth air and space) tetrahedron consists of 4 triangles cube (hexahedron) consists of 6 squares octahedron 8 triangles dodecahedron 12 pentagons icosahedron 20 triangles The hexagon didn't make it... possibly an infinite number of would assemble a sphere of infinite diameter, but this has never been included in any lists I've run across. In 4 dimensions, there are six convex 4-polytopes, called (polychorons), the smallest of which is called the pentatope, and is composes of 10 triangles, which can only be done in 4 dimensions, it can't be constructed under normal circumstances in our worlds. In 5, 6, 7, 8, 9, and 10 dimensions that are only 3 regular n-polytopes for each respectively... this may continue indefinitely but I don't know how to prove this, it's probably been done. If it does continue toward infinite dimensions that 2 and 3 dimensions are "special" and perhaps that is why we find ourselves in such a universe.

What is it called when 2 triangles can make a square?

When two triangles can form a square, it is often referred to as "triangular tiling" or "triangular decomposition." This concept is commonly seen in geometry, where right triangles can be arranged in such a way that their combined area equals that of a square. In particular, if the triangles are right triangles, they can be arranged to fit perfectly within the square's boundaries.

Can pentagon and square tessellate together?

No, a regular pentagon and a square cannot tessellate together. While squares can tessellate on their own, pentagons have angles that do not allow them to fit together with squares without leaving gaps. The internal angles of a regular pentagon are 108 degrees, while those of a square are 90 degrees, making it impossible to create a continuous tiling without overlaps or spaces.

What shape do you need to put with octagons to form a tiling pattern?

Assuming regular octagons, squares.

What are the five main of tessellation?

Tessellations are interesting and make for an interesting piece of art. There main types are normal tiling, monohedral tiling, isohedral tiling, penrose tiling, and voronoi tiling.

What are the five main types of tessellations?