Discrete geometry

Discrete geometry is a branch of mathematics that studies geometric objects and properties in a discrete setting, as opposed to continuous geometry. It focuses on structures that are made up of distinct, separate elements rather than continuous shapes or surfaces. This can include the study of points, lines, polygons, polyhedra, and more complex shapes, particularly in finite or countable settings.

Digital geometry is a field of study that deals with geometric objects and their representations in digital form, particularly in the context of computer graphics, image processing, and computer vision. It involves the mathematical analysis of shapes and structures that are represented as discrete pixels or voxels (in three dimensions) rather than continuous forms.

Azriel Rosenfeld is known as a pioneering figure in the field of computer science, particularly in image processing and computer vision. He has made significant contributions to the development of algorithms and methodologies for image analysis. Rosenfeld co-authored several influential publications and books in the areas of image processing and pattern recognition. Rosenfeld is particularly recognized for his work on the mathematical and algorithmic foundations that underlie various techniques in image processing, including edge detection, segmentation, and morphological operations.

A binary image is a digital image that consists of only two possible pixel values, typically represented as 0 and 1. In the context of image processing, these values usually correspond to two colors: one for the foreground (usually white or 1) and another for the background (usually black or 0).

Bresenham's line algorithm is an efficient algorithm used in computer graphics to determine which points in a grid or raster display should be plotted in order to form a straight line between two given points. It was developed by Jack Bresenham in 1962 and is particularly valued for being a simple, integer-based algorithm that runs quickly and does not require floating-point arithmetic.

Canberra distance is a metric used to measure the distance between two points in a multidimensional space, particularly for non-negative data. It is particularly useful in situations where the data may have different scales or when dealing with sparse data. The Canberra distance emphasizes the contributions of smaller values in the datasets, making it more sensitive to differences in low-value dimensions.

Closing in morphology refers to a process that involves the formation of morphemes by combining existing morphemes or modifying them. It typically describes how morphological structures can be completed or finalized, which can include morphological processes like affixation (adding prefixes or suffixes), compounding (combining two or more stems), or alternation (changing the form of a morpheme to express different grammatical categories).

A Controlled Image Base (CIB) is a digital representation of geographic and spatial information, specifically used in military and defense contexts. It provides a comprehensive and consistent framework for storing, managing, and distributing imagery and geospatial data. The CIB is designed to ensure that information about terrains, structures, and other features is readily accessible and can be effectively used for planning and operational purposes.

The Digital Differential Analyzer (DDA) is an algorithm used in computer graphics for line drawing. It is particularly important for rendering straight lines in raster graphics. The DDA algorithm is an incremental method that utilizes floating-point arithmetic to determine the points that lie on a straight line between two specified endpoints. ### Key Concepts of DDA 1.

A digital image is a representation of a two-dimensional image as a numerical grid of values. These values are typically organized in pixels, which are the smallest units that comprise the image. Each pixel contains information about the color or intensity at that specific point in the image. Digital images can be categorized into two main types: 1. **Raster Images (Bitmap Images)**: These images are made up of a grid of pixels, where each pixel represents a specific color.

Dilation is a fundamental operation in mathematical morphology, which is a branch of image processing that focuses on the shape and structure of features within images. Morphology uses a set of operations derived from set theory, lattice theory, topology, and random functions to analyze geometric structures in images. In the context of dilation, the process is applied to binary images (where pixels are represented as either foreground or background) or grayscale images.

Distance Transform is a technique used in image processing and computer vision that transforms a binary image into a distance map. The main objective of the distance transform is to calculate the distance of each pixel in the image to the nearest foreground pixel (typically the pixels belonging to a certain object or region of interest). ### Key Concepts: 1. **Binary Image**: A binary image consists of two pixel values, typically 0 (background) and 1 (foreground).

Erosion in the context of morphology refers to the process by which the structure or form of objects, particularly in the field of linguistics and morphology, undergoes gradual changes or reductions over time. In linguistics, morphology is the study of the internal structure of words, and erosion typically involves the simplification or loss of certain morphological features. For example, as languages evolve, complex word forms may become simplified.

The Euler operator is a concept from digital geometry, which deals with the study of geometric properties of shapes represented in a digital form, such as those found in computer graphics, image processing, and mathematical morphology. The Euler operator is used to calculate an important topological invariant known as the Euler characteristic of a digital object.

A gradually varied surface refers to a surface whose elevation or slope changes gradually over a certain distance. This term is often used in the context of hydrology, civil engineering, and fluid mechanics to describe surfaces like riverbeds, terrain, or other landscapes that exhibit subtle but consistent changes in height or depth. When analyzing flow over a gradually varied surface, engineers and scientists often focus on how these variations impact water movement, sediment transport, and other related processes.

The hit-or-miss transform is a morphological operation used in image processing and computer vision, particularly for shape matching and pattern recognition. It is a fundamental operation that allows one to detect specific shapes or patterns within a binary image. The hit-or-miss transform involves two sets: a structuring element (or template) and a binary image. The structuring element can be thought of as a defined shape or pattern that you want to detect in the image.

LCD crosstalk is a phenomenon that occurs in liquid crystal display (LCD) panels, particularly in those that use modern multi-layered technologies such as LCD screens with backlighting from LEDs. Crosstalk refers to the leakage of light from one pixel to adjacent pixels, which can cause blurring, ghosting, or double images in display content, especially during fast-moving scenes or when there are sharp edges between contrasting colors.

Mathematical morphology is a theoretical framework and a set of techniques for analyzing and processing geometric structures, often used in image analysis and computer vision. It was developed in the 1960s and 1970s, primarily by the mathematician Georges Matheron and his collaborator Jean Serra. The fundamental idea is to use set theory and lattice theory to study the shape and structure of objects in images.

The Midpoint Circle Algorithm is a graphical algorithm used to draw circles on computer screens or in raster graphics. It is particularly efficient because it uses only integer arithmetic, which helps in reducing computational overhead. The algorithm exploits the symmetry of circles to minimize the number of calculations needed. ### Key Concepts 1.

The term "morphological gradient" can refer to different concepts depending on the context in which it is used, but it primarily relates to two fields: morphology in linguistics and morphology in mathematical morphology (a branch of image processing). 1. **Linguistics**: In the context of linguistics, a morphological gradient refers to the variation in word forms or structures within a language. This can include how morphemes (the smallest meaningful units of language) combine and how this impacts meaning.

Morphological skeleton, often referred to simply as "skeletonization" in the context of image processing and computer vision, is a technique used in morphological image analysis. The purpose of morphological skeletons is to extract the essential structure of shapes in binary images (images composed of two colors, typically black and white) while reducing them to their simplest form.

A Nonogram, also known as a Picross or Griddler, is a logic puzzle that uses a grid to create a picture. The grid is accompanied by numeric clues that indicate the lengths of contiguous blocks of filled-in cells in each row and column. The objective of the puzzle is to fill in the grid according to these clues to reveal a hidden image.

Pick's Theorem provides a formula for determining the area of a simple lattice polygon (a polygon whose vertices are points with integer coordinates) based on the number of lattice points inside the polygon and on its boundary.

The term "Pixel" can refer to different things depending on the context. Here are some common meanings: 1. **In Digital Imaging**: A pixel (short for "picture element") is the smallest unit of a digital image that can be displayed or processed on a digital display system. Pixels combine to form images, and their resolution is often described in terms of width x height (e.g., 1920 x 1080 pixels).

Pixel aspect ratio (PAR) refers to the ratio of the width of a pixel to its height in a digital image or video. It is an important concept in digital imaging and video production because it affects how images and videos are displayed on different screens and devices. In a square pixel aspect ratio, the width and height of each pixel are equal (1:1), which is typical for most modern displays, cameras, and video formats.

Pruning in the context of mathematical morphology refers to a set of operations used in image analysis and processing, particularly for shape analysis. Morphology is a branch of mathematics that deals with the structure and form of objects, and it is often applied in computer vision and image processing to extract and analyze features of images. Pruning specifically involves reducing or simplifying the structure of shapes or objects in an image.

Raster graphics, also known as bitmap graphics, are images composed of a grid of individual pixels, where each pixel represents a specific color. This pixel-based approach means that raster images are resolution-dependent; their quality is determined by the number of pixels in the image (measured in resolution, such as DPI or PPI). Common formats for raster graphics include JPEG, PNG, GIF, and BMP.

Reeve tetrahedra refer to a particular type of geometric structure in the field of topology and computational geometry. Specifically, a Reeve tetrahedron is a tetrahedron that is formed through a specific type of triangulation of a polytope or a higher-dimensional manifold. The term is named after mathematician J. M. Reeve, who contributed to the understanding of geometric structures and properties in higher dimensions.

SPHARM-PDM stands for "Spherical Harmonic Parameterization and Shape Descriptor Model." It is a computational technique often used in the fields of medical imaging and computer graphics for analyzing and representing three-dimensional shapes, particularly biological structures. The approach uses spherical harmonics, a mathematical tool employed to represent functions on the sphere, to provide a compact and efficient way to describe the geometry of complex shapes.

A summed-area table (also known as an integral image) is a data structure used primarily in computer vision and image processing. It allows for rapid computation of the sum of pixel values in a rectangular subset of a grid or image. The key benefits of using a summed-area table include significantly reduced computation time and efficient querying for sums over image regions. ### How it Works: 1. **Construction**: A summed-area table is constructed from the original image by computing cumulative sums.

Taxicab geometry, also known as Manhattan geometry, is a form of geometry in which the distance between two points is calculated differently from the traditional Euclidean geometry. In Taxicab geometry, the distance between two points is the sum of the absolute differences of their coordinates, rather than the straight-line distance.

Thinning in the context of mathematical morphology is a morphological operation used primarily in image processing and computer vision. It is a technique that reduces the thickness of objects in a binary image while preserving their connectivity and shape. The goal of thinning is to simplify the representation of features in an image, often used for tasks like shape analysis, object recognition, or preprocessing for further analysis.

The top-hat transform is a mathematical morphology operation used in image processing and computer vision. It is particularly useful for enhancing features in images, such as bright spots or specific structures. The top-hat transform helps to extract small details from images, making it a valuable tool for various applications, including medical imaging, industrial inspection, and document analysis. ### Definition The top-hat transform can be defined as follows: 1. **Input Image:** Let \( f \) be the input image.

The term "topological skeleton" can refer to different concepts depending on the context in which it is used. Generally, it relates to the idea of simplifying or representing a complex structure in a way that captures its essential features while reducing unnecessary complexity.

Lattice points are points in a coordinate system whose coordinates are all integers. In a two-dimensional Cartesian coordinate system, a lattice point can be represented as \((x, y)\), where both \(x\) and \(y\) are integers. For example, the points \((1, 2)\), \((-3, 4)\), and \((0, 0)\) are all lattice points.

The Geometry of Numbers is a branch of number theory that studies the relationships between lattice points (points with integer coordinates) in Euclidean space and their geometric properties. It combines concepts from geometry, number theory, and algebra to address problems involving integers and their distribution within certain geometric shapes, particularly in relation to convex bodies.

The term "Bragg plane" is often associated with the field of crystallography and X-ray diffraction. It refers to a specific plane in a crystal lattice where constructive interference of X-rays occurs due to diffraction. When X-rays are scattered by the electron clouds of atoms in a crystal, the scattered waves can interfere with each other.

A Bravais lattice is a concept in crystallography that describes a specific arrangement of points in space, which represents the periodic repetition of a motif in three-dimensional space. It is defined by a set of discrete points that are arranged in a pattern that repeats at regular intervals, effectively forming the basis for the structure of a crystalline solid.

"Computing the Continuous Discretely" is a phrase commonly associated with the work and ideas of mathematician and computer scientist Steven Strogatz, particularly in the context of dynamical systems and complex systems. It highlights the interplay between continuous and discrete systems, illustrating how phenomena that are inherently continuous can be modeled, analyzed, or approximated using discrete computational methods.

The divisor summatory function, often denoted as \( \sum_{n \leq x} d(n) \), is a function that counts the total number of divisors of natural numbers up to \( x \). Specifically, \( d(n) \) represents the number of positive divisors of an integer \( n \).

Doignon's theorem is a result in the area of combinatorial geometry and specifically deals with the properties of finite sets of points in the Euclidean plane. It is sometimes described in the context of configuration spaces and combinatorial geometry. The theorem states that for any finite set of points in the plane, there exists a distinct set of lines such that the intersection of any two lines contains exactly one point from the original set.

A dot planimeter is an instrument used to measure the area of a two-dimensional shape or surface by tracing its perimeter. It is a type of planimeter that operates on the principle of dotting or marking points on the area being measured. The device typically consists of a tracing arm connected to a base and a measuring wheel.

A double lattice is a term that can refer to different concepts depending on the context in which it is used, particularly in mathematics, physics, and crystallography. 1. **Mathematics and Geometry**: In the context of lattices, a double lattice might refer to a structure formed from two interleaved or combined lattice structures.

The E8 lattice is an important and highly symmetrical structure in the field of mathematics, particularly in geometry and algebra. It is a type of lattice in eight-dimensional space and is one of the most studied examples in the theory of lattices due to its remarkable properties. ### Key Characteristics of the E8 Lattice: 1. **Definition**: - A lattice is a discrete group of points that is generated by linear combinations of basis vectors with integer coefficients.

Euclid's orchard is a mathematical concept that relates to the study of geometric configurations and properties, particularly in the context of number theory and combinatorial geometry. The term is not widely used in all mathematical contexts, but it can refer to a specific arrangement of points in a Euclidean space or an exploration of how to organize or distribute points according to certain rules or properties.

The Fokker periodicity block is a concept associated with certain types of mathematical models, particularly in statistical mechanics and quantum mechanics. It is named after the physicist A.D. Fokker, who contributed to the understanding of probabilistic distributions and their applications. In many-body systems, the term "periodicity" refers to the regular recurrence of certain properties in systems that exhibit periodic behavior, such as crystal lattices.

The Gauss circle problem is a classic problem in number theory and geometry that involves estimating the number of lattice points (points with integer coordinates) that lie within a circle of a certain radius centered at the origin in the Cartesian coordinate plane. More specifically, the problem asks how many integer points \((x, y)\) satisfy the inequality: \[ x^2 + y^2 \leq r^2 \] where \(r\) is the radius of the circle.

A hexagonal lattice is a type of arrangement of points (or lattice sites) in a two-dimensional plane where each point is positioned at the vertices of hexagons. This structure is characterized by the following key features: 1. **Geometry**: In a hexagonal lattice, each point has six nearest neighbors that are equidistant from it, forming a hexagonal shape. The angles between lines connecting a point to its neighbors are all 120 degrees.

An **integer lattice** is a discrete subset of Euclidean space formed by points whose coordinates are all integers.

In the context of group theory, a lattice is a partially ordered set (poset) that is closed under certain operations, specifically the operations of meet and join.

The Leech lattice is a specific type of lattice in 24-dimensional Euclidean space that has several remarkable properties. It was discovered by mathematician John Leech in the 1960s. Here are some key characteristics of the Leech lattice: 1. **Dimensions**: It exists in 24-dimensional space (R^24). 2. **Integral Lattice**: The Leech lattice is an integral lattice, meaning that its points (vectors) have coordinates that are all integers.

In mathematics, particularly in the field of topology and functional analysis, a Meyer set refers to a specific type of set associated with the theory of distributions and certain properties of functions in Sobolev spaces. More generally, the term can also refer to concepts in PDEs (partial differential equations) and harmonic analysis, but there isn't a universally accepted definition specifically for "Meyer set" across all mathematical disciplines.

The Niemeier lattices are a specific family of 24 even unimodular lattices in 24-dimensional space. They are named after the mathematician Hans Niemeier, who classified them in the 1970s. These lattices play an important role in various areas of mathematics, including number theory, geometry, and the theory of modular forms, as well as in theoretical physics, particularly in string theory and the study of orbifolds.

An oblique lattice refers to a specific type of two-dimensional lattice structure in crystallography and solid-state physics. In geometry, a lattice is a regular arrangement of points in space, and in the context of crystallography, it often describes the arrangement of atoms in a crystal. An oblique lattice is characterized by two non-orthogonal basis vectors that define a parallelogram in a two-dimensional space.

The Poisson summation formula is a powerful and essential result in analytic number theory and Fourier analysis, connecting sums of a function at integer points to sums of its Fourier transform. Specifically, it relates a sum over a lattice (for example, the integers) to a sum over the dual lattice.

The concept of the reciprocal lattice is fundamental in the field of solid-state physics and crystallography. It is a mathematical construct that helps in the analysis of wave phenomena in periodic structures, such as crystals. ### Definition: The reciprocal lattice is defined as a lattice in reciprocal space (momentum space), which is constructed from a given real space lattice (direct lattice). Each point in the reciprocal lattice corresponds to a unique set of wave vectors (k-vectors) associated with the periodic structure of the crystal.

A **rectangular lattice** is a type of lattice structure in a two-dimensional space that consists of points arranged in a grid-like pattern where the distances between neighboring points are constant in two perpendicular directions (typically referred to as the x- and y-directions). Each point in the lattice can be defined by the coordinates \((m, n)\), where \(m\) and \(n\) are integers.

A regular grid is a structured arrangement of points or cells that are uniformly spaced along one or more dimensions. This type of grid is characterized by its consistent intervals in both the x and y (and possibly z) directions, forming a predictable pattern. Regular grids are commonly used in various fields such as: 1. **Geography and GIS**: In geographical information systems (GIS), regular grids help in spatial analysis and representation of spatial data.

Schinzel's theorem is a result in number theory related to prime numbers and algebraic expressions. Specifically, it concerns the values of certain polynomial expressions and their ability to yield prime numbers for infinitely many integers. The theorem states that if \(P(x)\) is a polynomial with integer coefficients that takes on prime values for infinitely many integers \(x\), then it can be combined with another polynomial \(Q(x)\) to form a new polynomial that also takes prime values for infinitely many integers.

A square lattice is a type of two-dimensional geometric arrangement of points (or nodes) in which each point has four neighbors, located at equal distances from it, forming a square grid. In this arrangement, the points are positioned at the vertices of a square, with equal spacing between them in both the horizontal and vertical directions. Key characteristics of a square lattice include: 1. **Uniform Distance**: The distance between neighboring points is constant, which means that the lattice appears uniform across the entire plane.

An unimodular lattice is a type of mathematical structure that arises in the context of lattice theory and algebraic geometry, particularly in the study of quadratic forms and integer lattices. Here are the key characteristics and definitions associated with unimodular lattices: 1. **Lattice**: A lattice in Euclidean space is a discrete subgroup of that space generated by a set of basis vectors.

The mathematics of rigidity is a field that studies how structures maintain their shape and resist deformation under various forces. It encompasses a wide array of concepts and applications from geometry, topology, and structural engineering, focusing on both the theoretical and practical aspects of rigidity. ### Key Concepts in the Mathematics of Rigidity: 1. **Rigidity Theory**: This area investigates the conditions under which a geometric object (like a framework or structure) is rigid.

The Beckman–Quarles theorem is a result in the field of metric geometry pertaining to the nature of certain distance-preserving transformations. Specifically, it states that if \( f: \mathbb{R}^n \to \mathbb{R}^n \) is a function that preserves distances (i.e.

The Bricard octahedron is a type of self-intersecting polyhedron that is notable in the study of geometric structures and properties. Named after the French mathematician Georges Bricard, it is an example of a polyhedron with an unusual and complex structure. The Bricard octahedron has eight faces, all of which are congruent triangles. Unlike more regular polyhedra, it features intersections where the edges cross over one another.

Cauchy's theorem in geometry is a result concerning the properties of polygons, specifically convex polygons. The most well-known version pertains to the following statement: If two simple (non-intersecting) polygons are such that one can be continuously transformed into the other without self-intersection (while preserving the vertices and edges), then the two polygons have the same area.

The Cayley configuration space refers to an abstract mathematical concept primarily used in the study of algebraic geometry and topology, particularly in the context of algebraic groups and their representations. It is named after the mathematician Arthur Cayley. In general, the configuration space of a set of points (or particles) refers to the space of all possible positions these points can occupy, subject to certain constraints.

"Counting on Frameworks" typically refers to an approach in educational contexts, particularly in mathematics, where students build their understanding and skills by using structured frameworks or models for counting and number sense. This concept is often aimed at helping learners develop a solid foundation in numeracy through systematic counting strategies.

A flexible polyhedron is a type of polyhedron that can change its shape without altering the lengths of its edges. In other words, the vertices of a flexible polyhedron can move while keeping the distance between connected vertices constant, allowing the polyhedron to "flex" or deform. This characteristic distinguishes flexible polyhedra from rigid polyhedra, which cannot change shape without changing the lengths of their edges.

A **Laman graph** is a specific type of graph in the field of combinatorial geometry and rigidity theory. It is defined as follows: 1. **Vertices and Edges**: A Laman graph is a simple graph \( G \) with \( n \) vertices and \( m \) edges.

Parallel redrawing is a technique used in computer graphics and rendering that allows multiple parts of a scene or image to be redrawn simultaneously across different processing units, such as multiple CPU cores or GPU threads. This approach leverages the capabilities of modern hardware to improve rendering performance and efficiency. The basic idea of parallel redrawing is to divide the rendering task into smaller, independent workloads that can be processed concurrently.

A pseudotriangle is a geometric shape that resembles a triangle but does not necessarily meet all the criteria of a traditional triangle. The specific definition can vary depending on the context in which the term is used, such as in computational geometry or other mathematical fields. In some contexts, a pseudotriangle can refer to a polygon with three vertices that might not satisfy the requirements of having straight edges (i.e., it can contain curved segments) or other characteristics typically associated with standard triangles.

Steffen's polyhedron is a specific type of convex polyhedron that serves as a counterexample in geometric topology. It is notable for having a relatively simple construction but demonstrating interesting properties related to triangulations and face structures. More specifically, Steffen's polyhedron has the following key characteristics: 1. **Vertex Count**: It has 8 vertices. 2. **Edge Count**: It contains 24 edges.

Structural rigidity refers to the ability of a structure to maintain its shape and resist deformation when subjected to external forces or loads. It is an important property in engineering and architecture, as it impacts how buildings, bridges, and other structures respond to various types of stresses, including bending, twisting, and axial loads. Several factors influence structural rigidity, including: 1. **Material Properties:** The material used in a structure (e.g.

Packing problems are a class of optimization problems that involve arranging a set of items within a defined space in the most efficient way possible. These problems often arise in various fields such as operations research, logistics, manufacturing, computer science, and graph theory. The goal is usually to maximize the utilization of space, minimize waste, or achieve an optimal configuration based on certain criteria.

Bin packing is a type of combinatorial optimization problem that involves packing a set of items of varying sizes into a finite number of bins or containers in such a way that minimizes the number of bins used. The objective is to efficiently utilize space (or capacity) while ensuring that the items fit within the constraints of the bins. ### Key Concepts 1. **Items**: Each item has a specific size or weight. 2. **Bins**: Each bin has a maximum capacity that cannot be exceeded.

Circle packing is a mathematical concept that involves arranging circles within a given space, such that no two circles overlap and the arrangement satisfies certain criteria. The study of circle packing includes investigating how many circles of a given size can fit into a larger circle or how circles of different sizes can be arranged optimally.

Apollonian sphere packing is a fascinating and complex concept in geometry and number theory that involves the arrangement of spheres in three-dimensional space. The defining feature of Apollonian sphere packing is that it consists of an arrangement of spheres where each sphere is tangent to three others. Here’s a more detailed breakdown of the concept: ### Construction: 1. **Initial Configuration**: The process begins with three mutually tangent spheres. This creates a triangle of points where each sphere touches the others.

The Cutting Stock Problem is a classical optimization problem in operations research and production management. It deals with determining the most efficient way to cut raw materials (such as rolls of paper, metal, or wood) into smaller pieces or required lengths to meet specific demand. The goal is to minimize waste while fulfilling customer orders. ### Key Elements of the Problem: 1. **Raw Material:** Typically, a single large piece of material is used as a starting point (e.g., a large roll of paper).

Ellipsoid packing refers to the arrangement of ellipsoidal objects within a given volume in the most efficient way possible, often focusing on maximizing density—similar to how spheres can be packed. This concept arises in various fields, including mathematics, computer science, materials science, and physics. In three-dimensional space, the challenge of ellipsoid packing involves determining how to place ellipsoids (which can have different sizes and aspect ratios) to minimize the amount of unused space.

Hoffman's packing puzzle is a mathematical and geometric challenge that involves arranging a series of shapes in a way that they fit together without any gaps or overlaps. Specifically, it is often associated with packing an infinite number of circles, or spheres, in the most efficient way possible within a given space. The puzzle is named after the mathematician and computer scientist Charles Hoffman, who formulated it in 1992.

Parallel task scheduling refers to the method of organizing and managing multiple tasks or processes to be executed simultaneously on multiple processors or cores in a computing environment. This approach optimizes the use of computational resources and can significantly reduce the total execution time of a set of tasks compared to traditional sequential execution. Key concepts related to parallel task scheduling include: 1. **Task Decomposition**: Breaking a larger problem into smaller sub-tasks that can be solved independently and concurrently.

Polygon partition, often referred to as polygon triangulation in computational geometry, is the process of dividing a polygon into simpler components, typically triangles. This is useful for various applications in computer graphics, geographic information systems, and computational geometry because triangles are easier to work with for rendering and analysis.

Rectangle packing, also known as 2D packing or rectangular packing, is a combinatorial optimization problem where the goal is to pack a set of rectangles into a defined area (often referred to as a "bin" or "container") in the most efficient way. The objective can vary depending on the application, but common goals include minimizing the area of the container used, maximizing the number of rectangles that can be packed, or achieving a specific configuration.

The Slothouber–Graatsma puzzle is a type of mathematical or logical puzzle that is essentially a variation of a sliding puzzle often referred to as a "15 puzzle" or "sliding tile puzzle." In this puzzle, the objective is to slide tiles around on a grid to achieve a certain configuration, typically a numerical order or a specific pattern.

A smoothed octagon is a geometric shape that is derived from a regular octagon by rounding its corners. In terms of its definition and properties, it combines aspects of both polygonal and curved shapes. Here's how a smoothed octagon is typically characterized: 1. **Base Shape**: Start with a regular octagon, which has eight equal-length sides and eight equal angles (each measuring 135 degrees).

Sphere packing in a cube refers to the arrangement of non-overlapping spheres within a cube in such a way that optimizes the use of space. The goal is to maximize the number of spheres that can fit inside the cube while keeping them from intersecting. The most efficient known packing arrangement in three-dimensional space is called the face-centered cubic (FCC) or hexagonal close packing (HCP), which achieves a packing density of about 74.05%. This means that approximately 74.

Sphere packing is a mathematical concept that involves arranging spheres in a way that maximizes the amount of space filled by the spheres without any overlapping. In a three-dimensional space, the goal is to determine how many identical spheres can be packed into a larger sphere (or, sometimes, just in space) in the most efficient manner.

Square packing refers to the arrangement of objects, particularly in a two-dimensional space, where the items are packed into squares or rectangular grids in a way that optimizes space usage. This concept is commonly applied in various fields, including: 1. **Logistics and Shipping**: In warehousing and transportation, square packing involves organizing packages or pallets in a grid layout to maximize storage efficiency and minimize wasted space.

The strip packing problem is a classic optimization problem in the field of combinatorial optimization and computational geometry. The problem involves packing a set of items (usually rectangles) into a larger rectangular container, termed a "strip," with the objective of minimizing the height of the strip that is used. ### Problem Definition: 1. **Items**: You have a collection of rectangular items, each defined by its width and height.

Tetrahedron packing refers to the arrangement of tetrahedral shapes (the three-dimensional counterparts of triangles, with four triangular faces) in a space-efficient manner. This concept can be applied in various contexts, including materials science, chemistry, and mathematical optimization. In materials science, tetrahedron packing can describe the arrangement of atoms or molecules in a crystal lattice where the most efficient packing configurations can lead to the understanding of material properties.

"The Pursuit of Perfect Packing" refers to a mathematical and logistical challenge focused on the optimal arrangement of objects within a given space to maximize efficiency and minimize wasted volume. This topic intersects various fields, including geometry, packing problems, optimization, and even applications in computer science, engineering, and logistics. In the context of mathematics, perfect packing involves finding the best way to pack shapes or items into a defined space (like boxes or containers) without leaving empty gaps.

Tripod packing, also known as tripod positioning, is a technique used primarily in the context of managing respiratory distress. It involves a person leaning forward while supporting themselves on their arms, typically positioned on their knees or in a standing position. This stance allows the individual to open up their chest and diaphragm, facilitating easier breathing. This position is often seen in patients experiencing severe asthma attacks, chronic obstructive pulmonary disease (COPD) exacerbations, or other conditions that compromise respiratory function.

Ulam's packing conjecture is a hypothesis in the field of geometry and combinatorial mathematics, particularly concerning the arrangement of spheres in space. Formulated by mathematician Stanislaw Ulam, the conjecture posits that the densest packing of spheres (in three-dimensional space) occurs when the spheres are arranged in a face-centered cubic (FCC) lattice structure or equivalently in a hexagonal close packing (HCP) arrangement.

Discrete geometry is a branch of geometry that studies geometric objects and properties in a combinatorial or discrete context. It often involves finite sets of points, polygons, polyhedra, and other shapes, and focuses on their combinatorial and topological properties. Theorems in discrete geometry often relate to the arrangement, selection, or structure of these sets in specific ways.

Beck's theorem, in the context of geometry, generally refers to a result in the field of combinatorial geometry related to point sets and convex shapes. More specifically, it states that for any finite set of points in the plane, there exists a subset of those points that can be covered by a convex polygon of a certain size, where the size is influenced by the dimension of the space.

Carathéodory's theorem is a fundamental result in convex geometry that characterizes the representation of points in a convex set.

De Bruijn's theorem, named after the Dutch mathematician Nicolaas Govert de Bruijn, is primarily known in the context of combinatorics and graph theory. It refers to several important results, but the most widely recognized version is in relation to the properties of sequences and combinatorial structures.

The Erdős–Anning theorem is a result in the field of combinatorial number theory, particularly concerning sequences of integers and their properties regarding sums and subsets. Specifically, the theorem addresses the characterization of sequences that can avoid certain types of linear combinations.

The Erdős–Nagy theorem is a result in number theory that describes the conditions under which certain sequences can be generated by the marks made during a specific iterative process involving integers. More specifically, it concerns the distribution of sums of subsets of natural numbers. The theorem states that if \( A \) is a set of natural numbers, then the set of all finite sums formed by taking elements from \( A \) has certain properties related to density.

The Four-Vertex Theorem is a result in differential geometry and the study of curves. It states that for a simple, closed, smooth curve in the plane (which means a curve that does not intersect itself and is continuously differentiable), there are at least four distinct points at which the curvature of the curve attains a local maximum or minimum. To elaborate, curvature is a measure of how sharply a curve bends at a given point.

Helly's theorem is a result in combinatorial geometry that deals with the intersection of convex sets in Euclidean space. The theorem provides a condition for when the intersection of a collection of convex sets is non-empty.

Kirchberger's theorem pertains to the field of mathematics, specifically in the area of graph theory and combinatorial optimization. The theorem is often involved with properties of vertices and edges in graphs, particularly in relation to specific configurations or arrangements. However, it’s important to note that Kirchberger's theorem is not as widely known as some other mathematical theorems, so detailed and widely recognized references might be limited.

The Krein–Milman theorem is a fundamental result in convex analysis and functional analysis, particularly dealing with convex sets in the context of topological vector spaces. The theorem essentially provides a characterization of convex compact sets.

Monsky's theorem is a result in geometry related to the dissection of polygons. It states that it is impossible to dissect a square into a finite number of pieces, each of which is congruent to a given triangle with an odd area.

Radon's theorem is a result in convex geometry that deals with the intersection of convex sets. Specifically, it states that: **Radon's Theorem:** If a set of \( d + 2 \) points in \( \mathbb{R}^d \) is given, then it is possible to partition these points into two non-empty subsets such that the convex hulls (the smallest convex sets containing the points) of these two subsets intersect.

Tverberg's theorem is a result in combinatorial geometry that concerns the division of points in Euclidean space. It states that for any set of \( (r-1)(d+1) + 1 \) points in \( \mathbb{R}^d \), it is possible to partition these points into \( r \) groups such that the \( r \) groups share a common point in their convex hulls.

The Wallace–Bolyai–Gerwien theorem is a result in geometry related to the transformation of polygons. Specifically, it states that any two simple polygons of equal area can be dissected into a finite number of polygonal pieces that can be rearranged to form one another. The theorem has important implications in the study of geometric dissections, a topic that has intrigued mathematicians for centuries.

In geometry, triangulation refers to the process of dividing a geometric shape, such as a polygon, into triangles. This is often done to simplify calculations, especially in fields like computer graphics, spatial analysis, and geographic information systems (GIS). **Key points about triangulation in geometry:** 1. **Purpose:** Triangulation allows for easier computation of areas, volumes, and various properties of complex shapes since triangles are the simplest polygons.

An antiprism is a type of polyhedron that consists of two parallel polygonal bases connected by a band of triangular faces. It can be described as a generalized version of a prism, where instead of the bases being congruent and aligned, the bases are offset from each other and connected by equilateral triangles. Key features of an antiprism include: 1. **Polygonal Bases**: The two bases are identical polygons, such as triangles, squares, or pentagons.

An Apollonian network is a type of geometric network that is constructed using a recursive process based on the properties of triangular tiling. It begins with a single triangle, which is then subdivided into smaller triangles recursively. The network has a rich structure and exhibits fractal characteristics, making it interesting in the study of complex networks.

The Bowyer-Watson algorithm is a computational geometry algorithm used to incrementally construct a Delaunay triangulation of a set of points in a two-dimensional space. A Delaunay triangulation maximizes the minimum angle of the triangles formed, avoiding skinny triangles and ensuring better numerical stability for applications such as mesh generation and interpolation.

Constrained Delaunay triangulation (CDT) is a type of triangulation for a planar point set that respects certain constraints, particularly the inclusion of specified edges (or line segments) in the triangulation. This is an extension of the standard Delaunay triangulation, which is defined without any constraints.

Delaunay refinement is a computational geometry technique primarily used in the context of mesh generation. It aims to create a mesh composed of triangles (or tetrahedra in 3D) that satisfies certain optimality criteria, such as minimizing the maximum angle of the triangles (maximizing the minimum angle), and ensuring that the mesh conforms to specified geometric constraints of the underlying domain.

Delaunay triangulation is a geometric method for dividing a set of points into triangles such that no point is inside the circumcircle of any triangle in the triangulation. This property maximizes the minimum angle of the triangles, which helps avoid skinny triangles and is particularly useful in computational geometry and various applications including computer graphics, geographical information systems (GIS), and numerical simulations.

Fan triangulation is a method used in computational geometry, particularly in the field of computer graphics and geographic information systems. The process involves breaking down a polygon (usually a simple polygon) into a set of triangles, which can be more easily processed in various applications such as rendering or spatial analysis. The distinguishing feature of fan triangulation is that it typically starts from a single vertex (the "fan" vertex) and connects it to all other vertices of the polygon, forming a series of triangles.

Kinetic triangulation is a concept from computational geometry that deals with the dynamic problem of maintaining the properties of a triangulation of a set of points in motion. Specifically, it refers to the process of efficiently updating the triangulation structure as the points in the plane change their positions over time.

Minimum-weight triangulation (MWT) refers to the problem of dividing a simple polygon into triangles in such a way that the total weight of the edges used in the triangulation is minimized. The "weight" of an edge can be defined in various ways depending on the context, but it commonly relates to the length of the edge in geometric scenarios.

A **nonobtuse mesh** is a type of geometric mesh used primarily in finite element methods and computational geometry. In this context, a mesh is a collection of vertices, edges, and faces that defines a geometric shape or domain over which computations are performed. The term "nonobtuse" refers to the angles formed by the elements (usually triangles or tetrahedra) in the mesh.

Pitteway triangulation is a method used in mathematics and computer graphics for the triangulation of polyhedral surfaces, which involves breaking down a complex surface into simpler triangular components. This technique is particularly useful in computer graphics for rendering 3D models, as it simplifies the geometry and allows for easier manipulation and computation. The method typically involves defining a set of points on the surface and then systematically creating triangles that connect these points, ensuring that the entire surface is covered without overlaps or gaps.

Point-set triangulation is a computational geometry concept that involves subdividing a set of points into a collection of triangles, typically in a two-dimensional space. This method is essential for various applications in computer graphics, geographic information systems (GIS), finite element analysis, and mesh generation. In point-set triangulation, the key objectives are: 1. **Covering the Point Set**: The triangulation should cover all the points in the given set.

Polygon triangulation is the process of dividing a polygon into triangles, which are simpler geometric shapes. This is useful in various fields such as computer graphics, geographical information systems (GIS), and computational geometry because triangles are easier to work with for tasks like rendering, mesh generation, and mathematical computations.

Quasi-triangulation refers to a type of planar division that is similar to triangulation, but instead of dividing a region into triangles, it divides the region into a more generalized subdivision, which may include other polygonal shapes. This concept is relevant in computational geometry, where the goal is often to break down a complex shape into simpler components for analysis, representation, or processing.

Rotation distance, also known as **tree rotation distance**, is a concept from computational biology and bioinformatics that quantifies the minimum number of rotation operations required to transform one binary tree into another. A binary tree can be defined as a tree structure where each node has at most two children referred to as the left and right child. A rotation operation involves changing the structure of the tree without altering its nodes.

A simplicial complex is a mathematical structure used in algebraic topology and combinatorial mathematics to study spaces and their properties. It is a way of building up a geometric object from simpler building blocks called simplices. ### Definition of a Simplicial Complex A simplicial complex \( K \) is a set of simplices that satisfies two conditions: 1. **Non-emptiness**: The empty set is in \( K \).

A triangle mesh is a type of geometric representation commonly used in computer graphics, 3D modeling, and computational geometry. It consists of a collection of triangular faces that define a 3D shape or surface. Each triangle is typically defined by three vertices, which are points in 3D space, and the edges connecting these vertices.

A Triangulated Irregular Network (TIN) is a method used in geographic information systems (GIS) and computer graphics to represent a surface. It consists of a collection of triangles that are formed by connecting a set of irregularly spaced points (also known as vertices or nodes) in a way that creates a continuous representation of a surface, such as terrain elevation.

In topology, triangulation refers to the process of dividing a topological space into simpler pieces called simplices, specifically triangles (in two dimensions), tetrahedra (in three dimensions), or their higher-dimensional analogues. This technique is often employed in the study of geometric structures and algebraic topology.

Arrangement in the context of space partitioning refers to the way in which a geometric space is divided or partitioned based on a set of geometric objects, such as points, lines, or polygons. This partitioning can create distinct regions or cells within the space that can be analyzed or manipulated separately.

The term "arrangement of lines" can refer to various concepts depending on the context. Here are a few interpretations: 1. **Mathematics**: In geometry, the arrangement of lines could refer to the layout and positioning of lines in a plane, particularly how they intersect, are parallel, or are positioned relative to other geometric figures. This can involve discussions of line equations, slopes, and angles.

Bellman's lost in a forest problem is a classic problem in decision theory and optimal control, named after Richard Bellman, who developed dynamic programming. The problem illustrates how to formulate and solve problems involving uncertainty, where an agent must make a series of decisions in an unknown environment. ### The Problem Statement: The scenario involves a person who finds themselves lost in a forest. The person needs to determine which direction to go to find their way back to a known point (e.g.

The Big-line-big-clique conjecture is a concept in the field of combinatorics, more specifically in graph theory. It conjectures properties related to the structure and size of certain types of graphs, particularly concerning the relationships between cliques and line graphs. A clique in a graph is a subset of vertices such that every two distinct vertices in the subset are adjacent.

Borsuk's conjecture, proposed by Polish mathematician Karol Borsuk in 1933, asserts that any bounded, convex subset of Euclidean space \( \mathbb{R}^n \) can be partitioned into \( n + 1 \) or fewer subsets, each of which has a smaller diameter than the original set.

Carpenter's rule problem, often related to measuring and cutting materials in carpentry, involves practical challenges faced by carpenters when attempting to measure lengths accurately with a ruler that may have limited precision. One of the more classical interpretations of the Carpenter's rule problem involves determining how to cut a longer piece of wood into shorter lengths using only a limited-length ruler.

Centroidal Voronoi Tessellation (CVT) is a specific type of Voronoi tessellation where the sites of the Voronoi cells are chosen to be the centroids (centers of mass) of their respective cells. This idea combines the concepts of Voronoi diagrams and centroid calculations to optimize the placement of points in a given space, often leading to more evenly distributed and spatially balanced cell shapes.

Close-packing of equal spheres refers to the arrangement of spheres (or balls) in such a way that they occupy the maximum possible volume relative to the total volume of the space in which they are contained. This concept is particularly important in fields such as crystallography, materials science, and solid-state physics.

Combinatorial Geometry is a branch of mathematics that deals with the study of geometric objects and their combinatorial properties, often in a discrete setting. When we refer specifically to "Combinatorial Geometry in the Plane," we are primarily concerned with planar arrangements of points, lines, polygons, and other geometric figures, and how these arrangements relate to various combinatorial aspects.

The **connective constant** is a term used in statistical physics and combinatorics, particularly in the study of percolation theory and random walks on lattices. It quantifies the growth rate of connected clusters in a random graph or a lattice structure.

The Rado covering problem is a classic problem in combinatorics, particularly in the area of graph theory and set theory. The problem is named after mathematician Georgy Rado and deals with the concept of partitioning and covering subsets of sets. The problem can be stated in the following way: You are given a set \( S \), which is typically infinite, and a family of subsets of \( S \).

Discrete and Computational Geometry is a branch of mathematics and computer science that focuses on the study of geometric objects and their relationships, as well as the algorithms used to process and analyze these structures. It combines elements of combinatorial geometry, which deals with arrangements and properties of geometric objects, with computational geometry, which involves the development of algorithms to solve geometric problems.

The Disk Covering Problem is a combinatorial optimization problem related to covering a set of points in a multidimensional space using a minimal number of disks (or circles in 2D). The main goal is to determine the smallest number of disks of a given radius needed to cover all points in a specified area or space.

The Dissection Problem refers to a type of mathematical problem in geometry and combinatorial optimization where the goal is to dissect or cut a shape into a finite number of pieces that can be reassembled into another shape. This kind of problem often involves exploring how different shapes can be transformed into one another through geometric means.

Equidissection is a mathematical concept related to the idea of dividing shapes into pieces in such a way that the pieces can be rearranged to form another shape of equal area or volume. It involves partitioning a geometric figure into smaller pieces that can be reconfigured without changing their size, typically to demonstrate equivalence in area or volume between different figures. One of the popular contexts for discussing equidissection is in geometry, specifically in polygonal and polyhedral dissections.

The Erdős distinct distances problem, posed by the Hungarian mathematician Paul Erdős in 1946, is a question in combinatorial geometry that seeks to determine the minimum number of distinct distances between points in a given finite set in the plane. Specifically, the problem asks for the largest number of points \( n \) that can be placed in the plane such that the number of distinct distances between pairs of points is minimized.

The Erdős–Diophantine graph is a concept in graph theory that arises in connection with number theory and combinatorics, particularly focusing on the relationships defined by some Diophantine properties. In this setting, the vertices of the graph typically represent natural numbers or integers, and edges are drawn based on a specific Diophantine condition. The most common version of the Erdős–Diophantine graph considers pairs of integers that satisfy a particular equation or set of equations.

The Hadwiger Conjecture is a significant statement in combinatorial geometry that relates to the coloring of the plane with respect to convex sets, particularly focusing on the properties of regions defined by convex shapes.

Hinged dissection is a method in geometry that involves cutting a two-dimensional shape into pieces that can be folded or hinged around common points, allowing the pieces to reconfigure into another shape without overlapping. The concept is often illustrated using paper cutouts, where the cuts create "hinges" at specific points, enabling the pieces to pivot or swing into place. A classic example of hinged dissection is transforming a square into a triangle or vice versa.

The Honeycomb Conjecture is a mathematical statement regarding the most efficient way to partition a given area using shapes, specifically focusing on the arrangement of regular hexagons. The conjecture asserts that a regular hexagonal grid provides the most efficient way to divide a plane into regions of equal area with the least perimeter compared to any other shape.

An "integer triangle" typically refers to a triangle in which the lengths of all three sides are integers. For a triangle to exist with given side lengths, they must satisfy the triangle inequality theorem, which states that for any triangle with sides of lengths \( a \), \( b \), and \( c \): 1. \( a + b > c \) 2. \( a + c > b \) 3.

An **integrally convex set** refers to a special type of set in the context of integer programming and combinatorial optimization.

The term "isosceles set" does not appear to be a widely recognized term in mathematics or any specific field. However, it might be a misinterpretation or a confusion with the term "isosceles triangle," which refers to a triangle that has two sides of equal length.

A Kakeya set is a set of points in a Euclidean space (typically in two or higher dimensions) that has the property that a needle, or line segment, of unit length can be rotated freely within the set without leaving it. The classic example is the Kakeya set in the plane, which can be thought of as a bounded region that can contain a unit segment that can be rotated to cover all angles.

The Kepler conjecture is a famous problem in the field of discrete mathematics and geometry, specifically concerning the arrangement of spheres. It was proposed by the German mathematician Johannes Kepler in 1611. The conjecture states that no arrangement of spheres (or, more generally, circles or other three-dimensional shapes) can pack more densely than the face-centered cubic (FCC) packing or the hexagonal close packing (HCP).

The "kissing number" refers to the maximum number of non-overlapping spheres that can simultaneously touch another sphere of the same size in a given dimensional space. The concept can be applied in multiple dimensions, and the kissing number varies depending on the dimension. Here are some known kissing numbers: 1. **In 1 dimension**: The kissing number is **2**. A line segment (sphere in 1D) can touch two other line segments at its endpoints.

The Kobon triangle problem, also known as the "Kobon triangle," is a mathematical problem often discussed in the context of optimization and game theory. However, it seems there might be some confusion since the term "Kobon triangle problem" is not widely recognized in established mathematical literature up to my knowledge cutoff in October 2023.

Lebesgue's universal covering problem is a question in the field of topology, particularly concerning the properties of spaces that can be covered by certain kinds of collections of sets. Specifically, the problem asks whether every bounded measurable set in a Euclidean space can be covered by a countable union of sets of arbitrarily small Lebesgue measure.

The packing constant (or packing density) is a measure of how efficiently a shape can fill space when repeated. Different shapes have various packing constants based on how they can be arranged. Here is a list of some shapes with known packing constants: 1. **Circle**: - Packing Constant: \(\frac{\pi}{\sqrt{12}} \approx 0.9069\) for hexagonal packing 2.

The McMullen problem, posed by mathematician Curtis T. McMullen in the late 20th century, pertains to the study of hyperbolic 3-manifolds and their geometric structures. Specifically, it concerns the classification of certain types of 3-manifolds known as "hyperbolic 3-manifolds" and the conditions under which these manifolds can be represented as the complement of a knot in S³ (the 3-sphere).

Moser's worm problem is a thought experiment in mathematics and geometry, particularly in the field of topology and combinatorial geometry. It is named after the mathematician Jacob Moser, who posed it in the context of exploring geometric configurations and their properties. The problem can be outlined as follows: Imagine a straight worm of fixed length that can move through a two-dimensional plane.

The "Mountain Climbing Problem" typically refers to a type of optimization problem or search problem that can often be framed in the context of artificial intelligence, algorithms, or problem-solving techniques.

The Moving Sofa Problem is a classic problem in geometry and mathematical optimization. It involves determining the largest area of a two-dimensional shape (or "sofa") that can be maneuvered around a right-angled corner in a corridor. Specifically, the problem asks for the maximum area of a shape that can be moved around a 90-degree turn in a hallway, where the width of the hallway is fixed.

The Napkin Folding Problem is a classic problem in mathematics and combinatorial geometry, which involves determining the number of distinct ways to fold a napkin, typically represented as a two-dimensional sheet of paper. The goal is to explore how many unique configurations can be created through various folding techniques. The problem can be simplified into analyzing folds along a number of predefined lines, where each fold can change the orientation of the napkin.

In set theory and mathematics, an "opaque set" is not a standard or commonly used term. However, the concept of an opaque set might be used informally in certain contexts to refer to a set whose elements or the properties of which are not fully transparent or visible, or whose characteristics cannot be easily discerned. If you're encountering the term "opaque set" in a specific mathematical context, programming language, or another field, it may have a specialized meaning.

The orchard-planting problem is a problem in optimization typically found in operations research and mathematical programming. It involves the strategic placement of trees or plants in an orchard to maximize certain objectives while adhering to constraints. The problem can vary in its specifics, but it often includes considerations like: 1. **Maximizing Yield**: The primary goal is often to maximize the yield of fruits or nuts from the planted trees. This can depend on factors like tree density, spacing, and compatibility between different species.

Packing density, often referred to in contexts such as materials science, chemistry, and physics, is a measure of how densely a certain volume is filled with particles, such as atoms, molecules, or other small entities. It is typically expressed as a ratio or a percentage, quantifying the proportion of space occupied by the particles in comparison to the total available space.

Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles, named after the mathematician and physicist Roger Penrose, who studied these patterns in the 1970s. Unlike traditional tiling that can be periodically repeated, Penrose tilings cannot be exactly repeated in a regular pattern. They exhibit a form of symmetry that is both intricate and ordered, yet they do not repeat, which leads to fascinating mathematical and artistic properties.

Pinwheel tiling is a form of aperiodic tiling, which means it can cover a plane without repeating patterns while still being composed of simple geometric shapes. Specifically, pinwheel tiling uses a set of shapes known as "pinwheels" and is notable for its ability to create complex patterns that do not exhibit translational symmetry. The concept of pinwheel tiling was introduced by mathematician Robert Ammann in the 1970s.

A polycube is a three-dimensional geometric shape formed by joining several cubes together along their faces. These shapes can take various forms and configurations, depending on how the cubes are arranged. Polycubes can be considered a three-dimensional analog of polyominoes, which are shapes formed by connecting squares in two dimensions. Polycubes are often studied in mathematics and computer science for their properties and applications, including in fields like combinatorial geometry, topology, and even in puzzle design.

Quaquaversal tiling refers to a type of tiling pattern that exhibits a unique property of being the same regardless of the orientation from which it is viewed. The term "quaquaversal" is derived from a Latin term meaning "going in all directions," and in the context of tiling, it denotes a pattern that extends outward in multiple directions from a central point.

In graph theory, a **regular map** is a specific type of graph that satisfies certain symmetrical properties related to vertex and face structure.

Roberts's Triangle Theorem is a result in geometry concerning the relationship between the areas of certain triangles formed by points on the sides of a given triangle.

Sphere packing is the arrangement of spheres in a given space or volume in such a way that the spheres occupy the maximum possible volume without overlapping. It is a topic of interest in various fields such as mathematics, physics, and materials science. The most well-known packing configuration is the face-centered cubic (FCC) packing, which is one of the most efficient ways to pack spheres, achieving a maximum packing density of about 74%.

Sphere packing in a cylinder refers to the arrangement of spheres (or solid balls) within a cylindrical space in a way that maximizes the number of spheres that can fit inside the cylinder. This is a specific case of a more general problem in the field of discrete geometry and optimization, where the goal is to understand how to efficiently pack objects in given volumes.

"Squaring the square" refers to a mathematical problem in tiling, specifically involving the arrangement of squares within a square. The challenge is to subdivide a larger square into smaller squares, all of different sizes, such that there are no gaps or overlaps. The most famous solution to this problem was found by the mathematician Henry Dudeney in 1907. He created a square that was subdivided into 36 smaller squares, all of which were of distinct sizes.

A straight skeleton is a geometric construct that is generated from a polygon by tracing its edges and creating a new structure that reflects the shape of the original polygon. It is particularly significant in computational geometry and has applications in areas such as computer graphics, urban planning, and architecture. ### Definition To create a straight skeleton for a given polygon: 1. **Starting Point**: Begin with a simple polygon, which can be convex or concave but should not have holes.

Tarski's circle-squaring problem is a famous problem in the field of geometry and mathematics, proposed by the logician and mathematician Alfred Tarski in 1925. The problem involves the task of transforming a circle into a square (or vice versa) with the same area, using only a finite number of straightedge and compass constructions. Specifically, the question is whether it is possible to construct, with traditional geometric methods (i.e.

The Erdős Distance Problem is a classic problem in combinatorial geometry that concerns the maximum number of distinct distances that can be formed by a finite set of points in the plane. Specifically, the problem is named after the Hungarian mathematician Paul Erdős. The fundamental question can be stated as follows: Given a finite set of \( n \) points in the plane, what is the maximum number of distinct distances that can be formed between pairs of points in this set?

A Voronoi diagram is a mathematical structure that partitions a space into regions based on the distance to a specific set of points, called seed points or sites. Each region in a Voronoi diagram corresponds to one of the seed points, and every point within that region is closer to its associated seed point than to any other seed point.

A Weighted Voronoi Diagram is a variation of the standard Voronoi diagram that incorporates weights assigned to each point (or site) in the space. In a typical Voronoi diagram, the space is divided into regions based on the proximity to a set of points, where each point's region consists of all locations closer to that point than to any other.