Lecture

Division in Extreme and Mean Ratio: DEMR

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In course

MATH-124: Geometry for architects I

Ce cours entend exposer les fondements de la géométrie à un triple titre : 1/ de technique mathématique essentielle au processus de conception du projet, 2/ d'objet privilégié des logiciels de conception paramétrique 3/ de composante à part entière de l'h ...

Description

This lecture explores the concept of Division in Extreme and Mean Ratio (DEMR), introduced by Archytas and further developed by Euclide. DEMR is a geometric construction technique that involves dividing a line segment into two parts such that the ratio of the whole segment to the larger part is equal to the ratio of the larger part to the smaller part. The lecture delves into the historical evolution of DEMR, its applications in constructing regular polygons like the pentagon, and its connection to the Fibonacci sequence. It also discusses the significance of DEMR in art and architecture, highlighting its divine proportions and aesthetic implications.

Instructor

Bernard Cache

Official source

Ontological neighbourhood

Mathematics

Geometry: Euclidean geometry

Related concepts (32)

Geometry

Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

Regular polygon

In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line), if the edge length is fixed. These properties apply to all regular polygons, whether convex or star.

Star polygon

In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations on regular simple and star polygons. Branko Grünbaum identified two primary definitions used by Johannes Kepler, one being the regular star polygons with intersecting edges that don't generate new vertices, and the second being simple isotoxal concave polygons.

Differential geometry

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky.

Skew polygon

In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least four vertices. The interior surface (or area) of such a polygon is not uniquely defined. Skew infinite polygons (apeirogons) have vertices which are not all colinear. A zig-zag skew polygon or antiprismatic polygon has vertices which alternate on two parallel planes, and thus must be even-sided. Regular skew polygons in 3 dimensions (and regular skew apeirogons in two dimensions) are always zig-zag.

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