The Alexander horned sphere is a classic example in topology, specifically in the study of knot theory and manifold theory. It is constructed by taking a sphere and creating a complex embedding that demonstrates non-standard behavior in three-dimensional space. The construction of the Alexander horned sphere involves a series of increasingly complicated iterations that result in a space that is homeomorphic to the standard 2-sphere but is not nicely embedded in three-dimensional Euclidean space.

The term "lantern relation" is not widely recognized in most fields, and without additional context, it's challenging to determine its specific meaning. It could refer to a niche concept in a specialized area, or it could be a metaphorical or illustrative term in literature or art.

Geometric topology is a branch of mathematics that focuses on the properties of geometric structures on topological spaces. It combines elements of geometry and topology, investigating spaces that have a geometric structure and understanding how they can be deformed and manipulated. Here is a list of topics that are commonly studied within geometric topology: 1. **Smooth Manifolds**: - Differentiable structures - Tangent bundles - Morse theory 2.

The Blaschke selection theorem is a result in complex analysis and functional analysis concerning the behavior of sequences of Blaschke products, which are a type of analytic function associated with a sequence of points in the unit disk in the complex plane.

The Borromean rings are a set of three interlinked rings that are arranged in such a way that no two rings are directly linked together; instead, all three are interlinked with one another as a complete set. The key property of the Borromean rings is that if any one of the rings is removed, the remaining two rings will be unlinked, meaning they will not be entangled with each other.

The term "boundary parallel" can refer to different concepts depending on the context in which it is used. Generally, it relates to the idea of being aligned or closely associated with the boundaries of a particular system, area, or set of parameters. 1. **In Mathematics and Geometry**: Boundary parallel could describe lines, planes, or surfaces that run parallel to the edges or boundaries of a geometric shape or figure.

Boy's surface is a non-orientable surface that is an example of a mathematical structure in topology. It is a kind of 2-dimensional manifold that cannot be embedded in three-dimensional Euclidean space without self-intersections. Specifically, it can be constructed as a quotient of the 2-dimensional disk, and it can be visualized as a specific kind of "twisted" surface.

Hermann Brunn is not widely recognized in popular culture or historical context, so it's likely that you might be referring to a specific individual or a relatively obscure topic.

The term "crumpled cube" typically refers to a concept in the fields of materials science, mathematics, or physics, commonly associated with the study of shapes and structures. 1. **Materials Science**: In this context, a crumpled cube might study the deformation of materials, particularly how structures like a cube can be manipulated, folded, or crumpled to explore properties such as strength, stability, and energy absorption.

A Dehn twist is a fundamental concept in the field of topology, particularly in the study of surfaces and 3-manifolds. It is a type of homeomorphism that can be used to analyze the properties of surfaces and their mappings.

The Double Suspension Theorem is a concept in algebraic topology, particularly related to the behavior of suspensions in homotopy theory. The theorem provides a relationship between the suspension of a space and the suspension of built spaces from that space.

The Side-Approximation Theorem is a result in non-Euclidean geometry, particularly in the context of hyperbolic geometry. It relates to the conditions under which a triangle can be constructed in hyperbolic space given lengths of the sides.

In differential topology, the intersection form of a 4-manifold is an important algebraic invariant that captures information about how surfaces intersect within the manifold. Specifically, consider a smooth, closed, oriented 4-manifold \( M \). The intersection form is defined using the homology of \( M \).

The JTS Topology Suite (Java Topology Suite) is an open-source library designed for performing geometric operations on planar geometries. It is implemented in Java and follows the principles of the OGC (Open Geospatial Consortium) Simple Features Specification, which standardizes the representation and manipulation of spatial data.

Kirby calculus is a mathematical technique used in the field of low-dimensional topology, particularly in the study of 3-manifolds. It is named after Rob Kirby, who introduced this concept in a series of papers in the 1970s. The main focus of Kirby calculus is on the manipulation and understanding of 3-manifolds via the use of specific types of diagrams called Kirby diagrams or handlebody diagrams.

The Poincaré conjecture is a fundamental question in the field of topology, particularly in the study of three-dimensional spaces. Formulated by the French mathematician Henri Poincaré in 1904, the conjecture states that: **Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

Moise's Theorem, named after the mathematician Edwin Moise, is a result in the field of topology, specifically dealing with the characterization of certain types of surfaces. The theorem states that any triangle in Euclidean space can be decomposed into a finite number of pieces that can then be rearranged to form any other triangle, under a particular condition. In a more general sense, it also relates to the idea of "triangulation" of surfaces.

The Nielsen realization problem is a concept in the field of algebraic topology and group theory, specifically concerning the study of free groups and their automorphisms. More formally, it deals with the conditions under which a given group presentation can be realized as the fundamental group of a topological space, usually a certain type of surface or manifold.

The Geometry Center was a research and educational institution based in Minneapolis, Minnesota, that focused on the visualization of mathematical concepts, particularly in geometry and topology. Established in the late 1980s, the center aimed to promote the understanding of mathematical ideas through various means, including computer graphics, animations, and interactive software. It served as a hub for mathematicians, educators, and artists to collaborate on projects that highlighted the beauty and intricacies of geometry.

Interactive geometry software allows users to create and manipulate geometric constructions and models. These applications are commonly used in education for teaching geometry concepts, as well as by professionals in fields such as architecture and engineering. Here is a list of some popular interactive geometry software: 1. **GeoGebra** - A dynamic mathematics software that combines geometry, algebra, spreadsheets, graphing, statistics, and calculus.