A parallelogram is a four-sided polygon (quadrilateral) with two pairs of parallel sides. The opposite sides are not only parallel but also equal in length, and the opposite angles are equal. Some key properties of parallelograms include: 1. **Opposite Sides:** Both pairs of opposite sides are equal in length. 2. **Opposite Angles:** Both pairs of opposite angles are equal in measure.

Piecewise-circular curves are geometric constructions made up of multiple segments, where each segment can be represented as a circular arc. Instead of being a single continuous circular arc, the entire curve is comprised of several arcs that are connected at specific points, forming a continuous path. Each arc in a piecewise-circular curve can have different radii, and the points at which they connect can be chosen based on various criteria, such as smoothness, angle, or specific spatial constraints.

The 3-4-3-12 tiling refers to a specific way of covering a surface, often a plane or a geometric shape, using tiles or shapes that correspond to a specific arrangement or pattern. This term is often associated with a type of tiling that uses triangles and quadrilaterals.

The 3-4-6-12 tiling refers to a specific type of geometric tiling of the plane using polygons with angles that can create a regular tessellation pattern. In this case, the numbers 3, 4, 6, and 12 refer to the number of sides of the polygons used in the tiling: triangles (3 sides), squares (4 sides), hexagons (6 sides), and dodecagons (12 sides).

A Gaussian period is a mathematical concept that arises in number theory, specifically in the study of algebraic integers within cyclotomic fields. In particular, a Gaussian period is associated with the Gaussian integers, which are complex numbers of the form \( a + bi \), where \( a \) and \( b \) are integers and \( i \) is the imaginary unit.

A heptadecagon is a polygon with seventeen sides and seventeen angles. The term comes from the Greek word "hepta," meaning seven, and "deca," meaning ten, which when combined implies seventeen. In geometry, a regular heptadecagon has all sides and angles equal, and each internal angle measures approximately 156.47 degrees.

The mixtilinear incircle of a triangle is a special circle associated with a triangle, particularly in relation to its vertices and its incircle. For a given triangle \( ABC \), the mixtilinear incircle pertaining to a vertex, say \( A \), is the circle that is tangent to: 1. The incircle of triangle \( ABC \), 2. The arc \( BC \) of the circumcircle of triangle \( ABC \) that does not contain the vertex \( A \).

The term "Philo line" can refer to different concepts depending on the context, but it's most commonly associated with the study of religion, philosophy, or social theory. It may relate to the works of Philo of Alexandria, a Hellenistic Jewish philosopher whose ideas blended Jewish theology with Greek philosophy. In another context, "Philo" might refer to a specific concept or line of thought in philosophical discussions or literature.

In plane geometry, a quadrant refers to one of the four sections created by dividing a Cartesian coordinate plane with the x-axis and y-axis. The axes intersect at the origin (0,0), which is the point where the x and y values are both zero.

Thales's theorem is a fundamental result in geometry attributed to the ancient Greek mathematician Thales of Miletus.

"Squaring the circle" is a classic problem in geometry that involves constructing a square with the same area as a given circle using only a finite number of steps with a compass and straightedge. More formally, it requires finding a square whose area is equal to πr², where r is the radius of the circle. The problem has its origins in ancient Greece, where it was one of the three famous problems of antiquity, alongside duplicating the cube and trisecting an angle.

The Tienstra formula is primarily used in the field of physics, particularly in the study of fluid dynamics and thermodynamics, and is associated with calculating the properties of fluids in various conditions. However, in a more general scientific context, "Tienstra formula" may not be widely recognized or may refer to a specific application or derivation by a researcher named Tienstra.

A **Kan fibration** is a concept from category theory, particularly in the context of simplicial sets and homotopy theory. It generalizes the notion of a fibration in topological spaces to simplicial sets, allowing one to work with homotopical algebra. To understand Kan fibrations, we must first familiarize ourselves with simplicial sets.

In mathematics, specifically in the context of number theory, an "apotome" refers to a specific ratio or interval. The term originates from ancient Greek mathematics, where it was used to describe the difference between two musical tones or intervals. More precisely, the apotome is defined as the larger of two segments of the division of a musical whole.

In geometry, congruence refers to a relationship between two geometric figures in which they have the same shape and size. When two figures are congruent, one can be transformed into the other through a series of rigid motions, such as translations (shifts), rotations, and reflections, without any alteration in size or shape. Congruent figures can include various geometric objects, such as triangles, squares, circles, and polygons.

De Gua's theorem is a result in geometry that relates to right tetrahedra. It states that in a right tetrahedron (a four-faced solid where one of the faces is a right triangle), the square of the area of the face opposite the right angle (the right triangle) is equal to the sum of the squares of the areas of the other three triangular faces.

The term "double wedge" can refer to various concepts depending on the context. Here are a few interpretations: 1. **Mechanical Tool**: In mechanics or woodworking, a double wedge refers to a tool that consists of two wedge shapes often used for splitting or lifting materials. The design allows for more efficient force distribution.

In geometry, a half-space refers to one of the two regions into which a hyperplane (a flat subspace of one dimension less than the ambient space) divides the space.

In geometry, particularly in the study of figures in a plane or in space, the **homothetic center** refers to the point from which two or more geometric shapes are related through homothety (also known as a dilation). Homothety is a transformation that scales a figure by a certain factor from a fixed point, which is the homothetic center.

"On Conoids and Spheroids" is a notable work by the mathematician Giovanni Battista Venturi that was published in 1719. The treatise addresses the geometric properties of conoids and spheroids, which are forms generated by rotating curves around an axis. **Conoids** are surfaces generated by rotating a conic section (like a parabola) around an axis. They can exhibit interesting properties, such as the ability to create areas of uniform density when shaped correctly.