Slides
... (1) The class of valid ∀∃ sentences without function symbols is decidable. (2) The class of satisfiable ∃∀ sentences without function symbols is decidable. This results can extended to important cases where function symbols occur. ...
... (1) The class of valid ∀∃ sentences without function symbols is decidable. (2) The class of satisfiable ∃∀ sentences without function symbols is decidable. This results can extended to important cases where function symbols occur. ...
The parallel postulate, the other four and Relativity
... four).Since AB is common to ∞ Planes and only one Plane is passing through point M (Plane ABM from the three points A, B, M, then the Parallel Postulate is valid for all Spaces which have this common Plane, as Spherical, n-dimensional geometry Spaces. It was proved that it is a necessary logical con ...
... four).Since AB is common to ∞ Planes and only one Plane is passing through point M (Plane ABM from the three points A, B, M, then the Parallel Postulate is valid for all Spaces which have this common Plane, as Spherical, n-dimensional geometry Spaces. It was proved that it is a necessary logical con ...
Exploring triangles
... of the arrows, the quadrilaterals become more specialised. If two quadrilaterals in the diagram are linked by one or more arrows, the quadrilateral below is a special type of the one above it eg. an isosceles trapezium is a special type of trapezium. With a partner, decide whether the following stat ...
... of the arrows, the quadrilaterals become more specialised. If two quadrilaterals in the diagram are linked by one or more arrows, the quadrilateral below is a special type of the one above it eg. an isosceles trapezium is a special type of trapezium. With a partner, decide whether the following stat ...
Introduction to Geometry Review
... G-CO.9: Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistan ...
... G-CO.9: Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistan ...
chapter 9
... He assumed angles A and B to be right angles and sides AD and BC to be equal. His plan was to show that the angles C and D couldn’t both be obtuse or both be acute and hence are right angles. ...
... He assumed angles A and B to be right angles and sides AD and BC to be equal. His plan was to show that the angles C and D couldn’t both be obtuse or both be acute and hence are right angles. ...
3.1 The concept of parallelism
... Klein showed that there are three basically different types of geometry. In the Bolyai Lobachevsky type of geometry, straight lines have two infinitely distant points. In the Riemann type of spherical geometry, lines have no (or more precisely two imaginary) infinitely distant points. Euclidean geom ...
... Klein showed that there are three basically different types of geometry. In the Bolyai Lobachevsky type of geometry, straight lines have two infinitely distant points. In the Riemann type of spherical geometry, lines have no (or more precisely two imaginary) infinitely distant points. Euclidean geom ...
Isosceles Triangle (name the parts)
... Triangle "Side-Angle" Inequality: In a Triangle the LARGEST side is opposite the _______________________________ and the smallest side is opposite the _______________________________ ...
... Triangle "Side-Angle" Inequality: In a Triangle the LARGEST side is opposite the _______________________________ and the smallest side is opposite the _______________________________ ...
Euclidean Geometry and History of Non
... line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which asserts that there is exactly one line parallel to L passing through p. In elliptic geometry, there are no parallel lines at ...
... line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which asserts that there is exactly one line parallel to L passing through p. In elliptic geometry, there are no parallel lines at ...
Regular Tesselations in the Euclidean Plane, on the
... Introduction. The purpose of this article is to give an overview of the theory and results on tessellations of three types of Riemann surfaces: the Euclidean plane, the sphere and the hyperbolic plane. More general Riemann surfaces may be considered, for example toruses, but the theory is by far bet ...
... Introduction. The purpose of this article is to give an overview of the theory and results on tessellations of three types of Riemann surfaces: the Euclidean plane, the sphere and the hyperbolic plane. More general Riemann surfaces may be considered, for example toruses, but the theory is by far bet ...
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.