Dimensions is a
FrenchFrance , officially the French Republic , is a country located in Western Europe, with several overseas islands and territories located on other continents. Metropolitan France extends from the Mediterranean Sea to the English Channel and the North Sea, and from the Rhine to the Atlantic Ocean...
project that makes educational movies about
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, focusing on
spatial geometryIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions...
. It uses
POV-RayThe Persistence of Vision Raytracer, or POV-Ray, is a ray tracing program available for a variety of computer platforms. It was originally based on DKBTrace, written by David Kirk Buck and Aaron A. Collins. There are also influences from the earlier Polyray raytracer contributed by its author...
to render some of the animations, and the films are release under a Creative Commons licence.
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. It is the fifth largest of the eight planets in the solar system, and the largest of the terrestrial planets in the Solar System in terms of diameter, mass and density...
's coordinate system, and introduces the stereographic projectionIn geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it...
.
- Chapter 2: Dimension three discusses how two-dimensional beings would imagine three-dimensional objects.
- Chapters 3 and 4: The fourth dimension talk about four-dimensional polytope
In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions .When referring to an...
s (polychora), projecting the regular ones stereographically on the three-dimentional space.
- Chapters 5 and 6: Complex numbers are about the square root of negative numbers
A complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i
2 = −1...
, transformations, and fractalA fractal is "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
s.
- Chapters 7 and 8: Fibration show what a fibration
In mathematics, especially algebraic topology, a fibration is a surjective continuous mappingsatisfying the homotopy lifting property with respect to any space. Fiber bundles constitute important examples. In homotopy theory any mapping is 'as good as' a fibration — i.e...
is.
Discussion
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Dimensions is a
FrenchFrance , officially the French Republic , is a country located in Western Europe, with several overseas islands and territories located on other continents. Metropolitan France extends from the Mediterranean Sea to the English Channel and the North Sea, and from the Rhine to the Atlantic Ocean...
project that makes educational movies about
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, focusing on
spatial geometryIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions...
. It uses
POV-RayThe Persistence of Vision Raytracer, or POV-Ray, is a ray tracing program available for a variety of computer platforms. It was originally based on DKBTrace, written by David Kirk Buck and Aaron A. Collins. There are also influences from the earlier Polyray raytracer contributed by its author...
to render some of the animations, and the films are release under a Creative Commons licence.
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. It is the fifth largest of the eight planets in the solar system, and the largest of the terrestrial planets in the Solar System in terms of diameter, mass and density...
's coordinate system, and introduces the stereographic projectionIn geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it...
.
- Chapter 2: Dimension three discusses how two-dimensional beings would imagine three-dimensional objects.
- Chapters 3 and 4: The fourth dimension talk about four-dimensional polytope
In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions .When referring to an...
s (polychora), projecting the regular ones stereographically on the three-dimentional space.
- Chapters 5 and 6: Complex numbers are about the square root of negative numbers
A complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i
2 = −1...
, transformations, and fractalA fractal is "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
s.
- Chapters 7 and 8: Fibration show what a fibration
In mathematics, especially algebraic topology, a fibration is a surjective continuous mappingsatisfying the homotopy lifting property with respect to any space. Fiber bundles constitute important examples. In homotopy theory any mapping is 'as good as' a fibration — i.e...
is. Complex numbers are used again, and there are circleA circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the centre. The common distance of the points of a circle from its center is called its radius....
s and toriIn geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with and not touching the circle. Examples of tori include the surfaces of doughnuts and inner tubes. The solid contained by the surface is known as a toroid...
rotating and being transformed.
- Chapter 9: Proof emphasizes the importance of proofs
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
in mathematics, and proves the circle-conservationess of the stereographic projection as an example.
External links
The project's
websiteA website is a collection of related web pages, images, videos or other digital assets that are addressed with a common domain name or IP address in an Internet Protocol-based network...
is
http://www.dimensions-math.org/.