There are precisely three different classes of three-dimensional constant-curvature geometry: Euclidean, hyperbolic and elliptic geometry. The three geometries are all built on the same first four axioms, but each has a unique version of the fifth axiom, also known as the parallel postulate. The 1868Essay on an Interpretation of Non-Euclidean Geometryby Eugenio Beltrami (1835 – 1900) proved the logical consistency of the two Non-Euclidean geometries, hyperbolic and elliptical.
The parallel postulate is as follows for the corresponding geometries.
Euclidean geometry:Playfair’s version: “Given a lineland a pointPnot onl, there exists a unique linemthroughPthat is parallel tol.” Euclid’s version: “Suppose that a linelmeets two other linesmandnso that the sum of the interior angles on one side oflis less than 180°. Thenmandnintersect in a point on that side ofl.” These two versions are equivalent; though Playfair’s may be easier to conceive, Euclid’s is often useful for proofs.
Hyperbolic geometry:Given an arbitrary infinite lineland any pointPnot onl, there exist two or more distinct lines which pass throughPand are parallel tol.
Elliptic geometry:Given an arbitrary infinite lineland any pointPnot onl, there does not exist a line which passes throughPand is parallel tol.
“We shall therefore borrow all our Rules for the Finishing our Proportions, from the Musicians, who are the greatest Masters of this Sort of Numbers, and from those Things wherein Nature shows herself most excellent and compleat.” Leon Battista Alberti (1407-1472).
These pointers discuss triangles and their higher-dimensional generalizations (simplices). I am particularly interested in triangulation by which I mean partitioning regions into triangles, tetrahedra, or higher dimensional simplices, for various applications including finite element mesh generation and surface interpolation. (The other meaning of triangulation involves determining locations and distances from certain measurements.) For more material on the first type of triangulation, see the mesh generation section of Geometry in Action or the list of my own triangulation papers. For other kinds of partitions, see the page on dissection.
Plato considered geometry and number as the most reduced and essential, and therefore the ideal, philosophical language. But it is only by virtue of functioning at a certain ‘level’ of reality that geometry and number can become a vehicle for philosophic contemplation. Greek philosophy defined this notion of levels, so useful in our thinking, distinguishing the ‘typal‘ and the ‘archtypal‘. Following the indication given by Egyptian wall reliefs, which are laid out in three registers, an upper, a middle and a lower, we can define a third level, the ‘ectypal‘, situated between the archtypal and typal.
To see how these operate, let us take an example of a tangible thing, such as the bridle of a horse. This bridal can have a number of forms, materials, sizes, colours, uses, all of which are bridals. The bridal considered in this way, is typal; it is existing, diverse and variable. But on another level there is the idea or the form of the bridal, the guiding model of all bridals. This is an unmanifest, pure, formal idea and its level is ectypal. But yet above this there is an archtypal level which is that of the principal or power-activity, that is a process which the ectypal form and typal example of the bridal only represent. The archtypal is concerned with universal processes or dynamic patterns which can be considered independently of any structure or material form