Discusses such topics as the classical axiomatic systems of Euclid and Hilbert, axiom systems for three and four dimensional absolute geometry and Pieri's system based on rigid motions. Models, such as Taxicab Geometry, are used to illustrate theory.

The foundations of geometry and the non-Euclidean plane, by George E. Martin. Pp 509. $25·90. 1982. ISBN 3-540-90694-0 (Springer)

https://www.public.asu.edu/~pusunac/papers/jal.pdf

Axiomatizing geometric constructions

Neutral geometry is the geometry made possible with the first 28 theorems of Euclid’s Book 1—those results that do not rely on the parallel postulate. Hyperbolic geometry diverges from Euclidean geometry in that there are two parallels (asymptotic) to a given line through a given point and infinitely many other lines through the point that do not meet the given line (ultraparallel). In hyperbolic geometry, similar triangles are congruent; triangles have less than 180 degrees; there are no squares (regular four sided with 90 degree angles); and the area of a triangle is its defect (radian difference between π and the angle sum). As a consequence of this last remark, Bolyai (circa 1830) showed that one can square some circles (one uses a regular four sided polygon) in the hyperbolic plane using a ruler and compass [Gray 04,Curtis 90,Jagy 95]. Our aim here is to introduce origami constructions as a new method for doing geometry in the hyperbolic plane. We discuss the use of origami for making any ruler-compass construction and the possibilities for other constructions that are not ruler-compass constructions. In the first section, we give the alignments for folding in the hyperbolic plane. These are the analogues of the classical origami axioms in the plane discussed by Huzita and Justin. There is one additional alignment that is possible. Next, we discuss the relations between the alignments in the

/pdf/origami-alignments-and-constructions-in-the-hyperbolic-plane-1qyg7rd1b4.pdf

Origami Alignments and Constructions in the Hyperbolic Plane

Given two points inside a circle in the hyperbolic plane, we study the problem of finding an isosceles triangle inscribed in the circle so that the two points belong to distinct congruent sides By means of a reduction to the corresponding result in Euclidean geometry, we prove that this problem cannot generally be solved with hyperbolic ruler and compass In his treatise on optics, written in Arabic, the scientist and mathematician Abu Ali al-H  asan ibn al-Haytham (965‐1039) posed the problem of finding the light path between a source and an observer by way of a fixed spherical mirror, and gave a geometric solution for it The problem may have been formulated much earlier, by the great Greek mathematicians of the Hellenistic era, but no surviving testimony confirms this Thus it is fit that it carries al-H  asan ibn al-Haytham’s name, which was rendered as Alhazen in the Latin translation of his book — a document that played an important role in the development of modern science Alhazen recognized that the problem is essentially two-dimensional — the path must lie in a plane determined by the center of the sphere, the source and the observer His solution is long, in part because he is actually studying a more general problem; see [Sabra 1982] for details It is not a ruler-and-compass construction, as it requires an auxiliary hyperbola; in fact, apart from special cases, the problem turns out not to be solvable with ruler and compass alone, though it seems this was only proved some 50 years ago ([Elkin 1965]; see also [Riede 1989; Neumann 1998]) In this paper, we study the hyperbolic version of Alhazen’s problem and relate it to its classical Euclidean counterpart We use the following formulation of the problem: Given a circle (in the Euclidean or the hyperbolic plane) and two points A and B inside it, construct an inscribed, isosceles triangle with A on one equal leg and B on the other The isosceles condition is equivalent to the condition that the two legs meet at equal angles the diameter of the circle that goes through their common vertex, so this is Alhazen’s problem all right (One can also imagine a round billiard table

/pdf/alhazen-s-hyperbolic-billiard-problem-fir208utp6.pdf

Alhazen’s hyperbolic billiard problem

The Euclidean classical problem ofDuplicating the Cube is investigated for the first time inhyperbolic geometry, and its partial solution presented. The other two associated problems ofTrisecting an Angle andSquaring the Circle in the hyperbolic plane were solved classically, but a few additions to these solutions are presented, along with some miscellaneous notes about straightedge and compass constructions in the hyperbolic plane.

Robert R. Curtis

Papers

Duplicating the cube and other notes on constructions in the hyperbolic plane