Question

Two-dimensional lattices with three- or four-sided symmetries are possible, but there is none with a five-sided symmetry. To see why, consider the following: A piece of paper can be cut into identical equilateral triangles or squares with no excess. Prove that this is not the case for equilateral pentagons.

Short answer

The main reason why one can't create a lattice using identical equilateral pentagons is that the sum of the angles at any point in a two-dimensional tiling must be equal to 360 degrees to fit without gaps. The internal angles of a pentagon are 108 degrees, which does not divide evenly into 360, leading to gaps in the tiling.

Step-by-step solution

Understanding the Problem

This problem is about proving that it is impossible to fill a plane without any 'gaps' using only equilateral pentagons. A lattice is a regular, periodic configuration of points, particles, or objects throughout an area or a space. It's important to understand that regular triangles and squares, three and four sided polygons, can tile a plane without any extra space or overlap.

Understanding Symmetry in Tiling

A regular tiling of the plane by a specific type of shape is a covering of the infinite plane by congruent copies of the shape, with no overlaps and no gaps. The symmetry type of a tiling is the symmetry group of its pattern of identical tiles, which considers not only translations, but also rotations and reflections. In the case of regular triangles and squares, a plane can indeed be filled without any gaps, because the internal angles of these shapes are exact divisors of 360 degrees.

Why Pentagons Don't Work

An equilateral pentagon does not tessellate the plane. If we look at the angles in an equilateral and equiangular pentagon, each of the interior angles is 108 degrees. However, the sum of the angles at any point in a 2D tiling must be equal to 360 degrees for the shapes to fit without any gaps. Since 108 doesn't divide evenly into 360, an attempt to tile the plane with pentagons will result in gaps.

Key concepts

Symmetry in Tiling

The beauty and complexity of tiling patterns lie in their symmetry. Symmetry in tiling refers to the regular and repetitive pattern by which a shape or set of shapes cover a plane without any gaps or overlaps. Each tiling system can be characterized by its symmetry group, representing all the motions that can be applied to the tiling which will leave it looking the same.
In the realm of geometry, the term symmetry is associated with operations known as 'isometries' which include reflections, rotations, translations, and glide reflections. These isometries are the movements to achieve identical placements within the tiling pattern. For instance, a square can be rotated 90 degrees, 180 degrees, or 270 degrees around its center, as well as reflected across its diagonals and sides, and the pattern will maintain its symmetric appearance. This inherent symmetry allows for tiles like squares or regular triangles to fill a plane entirely, resulting in a beautiful mosaic of predictable and contiguous patterns.
As you might imagine, the more complex the shape, the more complex the symmetry—if it even exists. In the case of equilateral pentagons, the lack of 'fit' when tiling a plane suggests a break in such symmetry, unveiling an intrinsic geometric limitation of the shape.

Lattice Structures

In geometry, a lattice is a highly ordered structure of points that extends infinitely in two or more directions. These structures form the backbone for a variety of patterns and designs seen in tiling. Imagine a grid where points are distributed at uniform intervals, this provides an even framework upon which shapes can be placed to cover an entire plane—a process known as tessellation.
Lattice structures are deeply interconnected with symmetry. When we use basic shapes, like equilateral triangles or squares, these shapes' angles perfectly match the grid-like framework of the lattice and tessellate seamlessly. Each shape essentially clicks into place without wasting any space or creating any overlaps. Consequently, these aligned structures illustrate a perfect balance and repetitive uniformity, easily visualized by the grid of a lattice.
Such a neatly arranged lattice is not just a trait of theoretical geometry; it's also visible in many natural and human-made objects, from the atomic lattices of crystals to the urban grids of our cities. But as straightforward as lattices may seem, they cannot incorporate shapes like equilateral pentagons, as the pentagons won't fit neatly within the grid's confines.

Equilateral Pentagons

An equilateral pentagon is a five-sided polygon with all sides of equal length. Unlike shapes with three or four sides, the equilateral pentagon poses a unique challenge in the world of tiling. Its interior angles, each measuring 108 degrees, do not nicely divide into the 360-degree rotational symmetry required for a seamless tiling pattern.
If we try to fit equilateral pentagons together at a point, we find that we can only fit three pentagons around a single vertex before we exceed 360 degrees (3 * 108 = 324). That leaves an unavoidable gap of 36 degrees—a problematic shortfall indicating that a tiling of the plane with only equilateral pentagons is impossible without breaking symmetry or leaving gaps. This curious property of pentagons stands in stark contrast to the ease with which shapes like squares or equilateral triangles tessellate, each neatly fitting into the lattice framework like pieces of a puzzle.
Despite their challenging nature, pentagons have inspired numerous geometrical explorations and artistic projects, demonstrating that the universe of shapes is as vast and varied as it is fascinating.

Plane Tessellation

Plane tessellation, or tiling, is a captivating topic in both mathematics and art. It is the process of covering a plane with a pattern of flat shapes so that there are no overlaps and no gaps. This process requires the repeating shapes to fit together precisely like a jigsaw puzzle.
The noble goal of tessellation is to achieve a pattern that could, in theory, extend infinitely in all directions. Simple shapes that have internal angles that are divisors of 360 degrees can tessellate the plane easily. For example, the internal angles of equilateral triangles are 60 degrees and squares are 90 degrees, both of which can perfectly tile a plane by repeating the pattern over and over.
By extension, tessellation not only demonstrates mathematical concepts but also showcases the inherent artistic beauty through patterns seen everywhere from historical architecture to modern graphic designs. It spans cultures and ages, a testament to both human creativity and the elegance of mathematical laws.