Dimensions is a French project that makes educational movies about

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, focusing on

spatial geometry Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...

. It uses

POV-Ray The Persistence of Vision Ray Tracer, most commonly acronymed as POV-Ray, is a cross-platform ray-tracing program that generates images from a text-based scene description. It was originally based on DKBTrace, written by David Kirk Buck and Aaro ...

to render some of the animations, and the films are released under a

Creative Commons licence A Creative Commons (CC) license is one of several public copyright licenses that enable the free distribution of an otherwise copyrighted "work".A "work" is any creative material made by a person. A painting, a graphic, a book, a song/lyrics ...

. The film is separated in nine chapters, which follow this plot: * Chapter 1: Dimension two explains

Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...

's coordinate system, and introduces the

stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...

. * Chapter 2: Dimension three discusses how two-dimensional beings would imagine three-dimensional objects. * Chapters 3 and 4: The fourth dimension talks about

four-dimensional A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...

polytope In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...

s (''polychora''), projecting the regular ones stereographically on the three-dimensional space. * Chapters 5 and 6: Complex numbers are about the

square root of negative numbers An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . Fo ...

transformation Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Trans ...

s, and

fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...

s. * Chapters 7 and 8: Fibration show what a

fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all map ...

is. Complex numbers are used again, and there are

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

s and tori rotating and being transformed. * Chapter 9: Proof emphasizes the importance of

proofs Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...

in mathematics, and proves the circle-conservationess of the stereographic projection as an example. They are available for download in several languages.

References

External links

* Animated film series {{topology-stub