Introduction to the Obscure Topic of Tessellations of Hyperbolas: A Comprehensive Guide
Tessellations of hyperbolas are a fascinating and often overlooked branch of mathematics. This article aims to provide a comprehensive guide to tessellations of hyperbolas, including their definition, properties, and applications.
What are Tessellations of Hyperbolas?
A tessellation of a hyperbola is a system of hyperbolas that distinguishably covers two-dimensional space with no repeating space between them. In other words, when one hyperbola is drawn in the plane, its reflection through an axis must intersect another hyperbola to form a tessellation. The axes of the hyperbolas must also be disjoint and of different orientations.
There are two types of hyperbolas: the upper and lower hyperbolas. The upper hyperbola has two branches, while the lower has none. The distance between the foci of a hyperbola is called the focal length, while the distance between the centers of its branches is called the asymptotic distance. The vertices of a hyperbola are located at the intersection of the axes, while the center is located at the origin.
Properties of Tessellations of Hyperbolas
Tessellations of hyperbolas have several interesting properties that make them useful in various applications.
Translational Symmetry
Tessellations of hyperbolas have translational symmetry, meaning that they are the same after a rigid translation. The translations must be vertical or horizontal and of arbitrary size, but not diagonal.
Rotational Symmetry
Tessellations of hyperbolas also have rotational symmetry, meaning that they are the same after rotating from any angle around a specified center point. The angle of rotation must be a multiple of 60 degrees.
Reflexional Symmetry
Tessellations of hyperbolas have reflexional symmetry, meaning that they are the same after reflecting through any axis. The axis of reflection must be vertical or horizontal and pass through a vertex of the hyperbola.
Grid Form
Tessellations of hyperbolas can be represented as grids of hyperbolas, where each hyperbola is represented as a segment of an edge. The grid is usually a square or a rectangle, with all the sides of a given size and angle.
Applications of Tessellations of Hyperbolas
Tessellations of hyperbolas have several real-world applications in various fields, including architecture, physics, and engineering.
Architecture
Tessellations of hyperbolas are used in architecture to create beautiful, geometric shapes that are both functional and aesthetic. The most famous example of a herringbone tessellation of hyperbolas is the mosaic floor in the Chorus Church in Visoko, Bosnia.
Another example of a tessellation of hyperbolas in architecture is the Oakland County Courthouse in Michigan, which features a complex system of hyperbolic arches and domes.
Physics
Tessellations of hyperbolas are also used in physics to describe the behavior of particles in high-energy systems. One example of this is the behavior of electron orbits in atoms, which are described using hyperbolic orbits.
Another example is the behavior of cosmic rays, which are broken, uncharged particles that travel through space and interact with matter on Earth. The trajectory of a cosmic ray can be described using a hyperbolic equation.
Engineering
Tessellations of hyperbolas are used in engineering to design structures that are both strong and lightweight. One example of this is the design of bridges and arches, which use hyperbolic shapes to distribute stress and maximize the span between supports.
Another example is the design of satellites and spacecraft, which use hyperbolic trajectories to optimize their path through space.
Types of Tessellations of Hyperbolas
There are two main types of tessellations of hyperbolas: the horizontal tessellation and the vertical tessellation. These two types are distinguished by the orientation of the hyperbolas and the direction of the lines connecting the centers of the hyperbolas.
Horizontal Tessellations
In a horizontal tessellation of hyperbolas, the hyperbolas are oriented horizontally, with the axes resting on a vertical line. The lines connecting the centers of the hyperbolas are oriented in the same direction as the hyperbolas, but are parallel to each other.
There are three types of horizontal tessellations of hyperbolas: the regular square tessellation, the regular hexagonal tessellation, and the regular tessellation with a hexagonal base and a circular branching system. The regular square tessellation is composed of two square grids of hyperbolas, connected by lines that cross the grid diagonals. The regular hexagonal tessellation is composed of three hexagonal grids of hyperbolas, connected by lines that cross the grid diagonals. The regular tessellation with a hexagonal base and a circular branching system is composed of one hexagonal and two circular grids of hyperbolas, connected by lines that cross the grid diagonals.
Vertical Tessellations
In a vertical tessellation of hyperbolas, the hyperbolas are oriented vertically, with the axes resting on a horizontal line. The lines connecting the centers of the hyperbolas are oriented in the same direction as the hyperbolas, but are parallel to each other.
There are three types of vertical tessellations of hyperbolas: the regular parallelogram tessellation, the regular scalene tessellation, and the regular trapezoid tessellation. The regular parallelogram tessellation is composed of two parallelogram grids of hyperbolas, connected by lines that cross the grid are diagonals. The regular scalene tessellation is composed of two scalene grids of hyperbolas, connected by lines that cross the grid diagonals. The regular trapezoid tessellation is composed of one trapezoidal and two cubic grids of hyperbolas, connected by lines that cross the grid diagonals.
Conclusion
Tessellations of hyperbolas are a fascinating and often overlooked branch of mathematics with a wide range of applications in various fields. From architecture to physics to engineering, tessellations of hyperbolas have shown that simple shapes can be used to create beautiful and functional systems. As such, it is important to understand the properties and applications of tessellations of hyperbolas in order to fully appreciate the power of this branch of math.