[edit] Construction of the Lebesgue measure

... An early paper by Benoit Mandelbrot entitled How Long Is the Coast of Britain? Statistical SelfSimilarity and Fractional Dimension and subsequent work by other authors have claimed that the Hausdorff dimension of many coastlines can be estimated. Their results have varied from 1.02 for the coastline ...

... An early paper by Benoit Mandelbrot entitled How Long Is the Coast of Britain? Statistical SelfSimilarity and Fractional Dimension and subsequent work by other authors have claimed that the Hausdorff dimension of many coastlines can be estimated. Their results have varied from 1.02 for the coastline ...

Ratios in Similar Polygons Ratios in Similar Polygons

... Writing a similarity statement is like writing a congruence statement—be sure to list corresponding vertices in the same order. ...

... Writing a similarity statement is like writing a congruence statement—be sure to list corresponding vertices in the same order. ...

Basic Geometry - Congruence Similar and Angle Relationships

... OBJECTIVE: Students will understand and apply basic geometry skills to similar and congruent figures, as well as, angle relationships. Introduction: Two geometric figures that have the same size and shape are said to be congruent. Congruence is one of the most important concepts in basic geometry. C ...

... OBJECTIVE: Students will understand and apply basic geometry skills to similar and congruent figures, as well as, angle relationships. Introduction: Two geometric figures that have the same size and shape are said to be congruent. Congruence is one of the most important concepts in basic geometry. C ...

Non-Euclidean Geometry Topics to Accompany Euclidean and

... In 1829 a Russian mathematics professor named Nikolai Lobachevsky from the University of Kasan published “On the Principles of Geometry” in the Kasan Bulletin. In this article, he described a geometry in which more than one parallel to a given line may be drawn through a point not on the line. He fo ...

... In 1829 a Russian mathematics professor named Nikolai Lobachevsky from the University of Kasan published “On the Principles of Geometry” in the Kasan Bulletin. In this article, he described a geometry in which more than one parallel to a given line may be drawn through a point not on the line. He fo ...

G5-5-Indirect Proof

... So far you have written proofs using direct reasoning. You began with a true hypothesis and built a logical argument to show that a conclusion was true. In an indirect proof, you begin by assuming that the conclusion is false. Then you show that this assumption leads to a contradiction. This type of ...

... So far you have written proofs using direct reasoning. You began with a true hypothesis and built a logical argument to show that a conclusion was true. In an indirect proof, you begin by assuming that the conclusion is false. Then you show that this assumption leads to a contradiction. This type of ...

Ab-initio construction of some crystalline 3D Euclidean networks

... The symmetries of the map reveal the underlying hyperbolic orbifold. Its most compact form contains a single copy of the orbifold. The universal cover in H2 consists of infinitely many copies of the orbifold, generated by repeated reflections in all (2D mirrors) boundary arcs. The orbifold pertainin ...

... The symmetries of the map reveal the underlying hyperbolic orbifold. Its most compact form contains a single copy of the orbifold. The universal cover in H2 consists of infinitely many copies of the orbifold, generated by repeated reflections in all (2D mirrors) boundary arcs. The orbifold pertainin ...

# Space

Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework.Debates concerning the nature, essence and the mode of existence of space date back to antiquity; namely, to treatises like the Timaeus of Plato, or Socrates in his reflections on what the Greeks called khôra (i.e. ""space""), or in the Physics of Aristotle (Book IV, Delta) in the definition of topos (i.e. place), or in the later ""geometrical conception of place"" as ""space qua extension"" in the Discourse on Place (Qawl fi al-Makan) of the 11th-century Arab polymath Alhazen. Many of these classical philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, particularly during the early development of classical mechanics. In Isaac Newton's view, space was absolute—in the sense that it existed permanently and independently of whether there was any matter in the space. Other natural philosophers, notably Gottfried Leibniz, thought instead that space was in fact a collection of relations between objects, given by their distance and direction from one another. In the 18th century, the philosopher and theologian George Berkeley attempted to refute the ""visibility of spatial depth"" in his Essay Towards a New Theory of Vision. Later, the metaphysician Immanuel Kant said that neither space nor time can be empirically perceived—they are elements of a systematic framework that humans use to structure all experiences. Kant referred to ""space"" in his Critique of Pure Reason as being a subjective ""pure a priori form of intuition"", hence it is an unavoidable contribution of our human faculties.In the 19th and 20th centuries mathematicians began to examine geometries that are not Euclidean, in which space can be said to be curved, rather than flat. According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a better model for the shape of space.