Squaring the Circle and Doubling the Cube in Space-Time
- Art, PhysicsThe Mathematics Enthusiast
Squaring the Circle is a famous geometry problem going all the way back to the ancient Greeks. It is the great quest of constructing a square with the same area as a circle using a compass and…
Squaring the Circle in Elliptic Geometry
Constructing a regular quadrilateral (square) and circle of equal area was proved impossible in Euclidean geometry in 1882. Hyperbolic geometry, however, allows this construction. In this article, we…
Squaring The Circle In The Hyperbolic Disk
Bolyai ended his 1832 introduction to non-Euclidean geometry with a strategy for constructing regular quadrilaterals (squares) and circles of the same area. In this article, we provide the steps for…
Hypercomputation: Fantasy or Reality? A Position Paper
- PhilosophyParallel Process. Lett.
These arguments against hypercomputation are not unwavering as they seem to be and here I explain why.
From Hilbert's Axioms to Circle-squaring in the Hyperbolic Plane
This thesis is based on M. J. Greenberg's article "Old and New Results in the Foundation of Elementary Plane Euclidean and Non-Euclidean Geometries" (American Mathematical Monthly , Vol 117, No 3 pp.…
Origami Alignments and Constructions in the Hyperbolic Plane
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…
Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries
- MathematicsAm. Math. Mon.
This survey highlights some foundational history and some interesting recent discoveries that deserve to be better known, such as the hierarchies of axiom systems, Aristotle's axiom as a “missing link,” Bolyai’s discovery of the relationship of “circle-squaring” in a hyperbolic plane to Fermat primes, the undecidability, incompleteness, and consistency of elementary Euclidean geometry.
SHOWING 1-6 OF 6 REFERENCES
The Book of Prime Number Records
1. How Many Prime Numbers Are There?.- I. Euclid's Proof.- II. Kummer's Proof.- III. Polya's Proof.- IV. Euler's Proof.- V. Thue's Proof.- VI. Two-and-a-Half Forgotten Proofs.- A. Perott's Proof.- B.…
The foundations of geometry and the non-Euclidean plane
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…
1. The historical development of non-Euclidean geometry 2. Real projective geometry 3. Real projective geometry: polarities conics and quadrics 4. Homogeneous coordinates 5. Elliptic geometry in one…
The new book of prime number records
1 How Many Prime Numbers Are There?.- I. Euclid's Proof.- II. Goldbach Did It Too!.- III. Euler's Proof.- IV. Thue's Proof.- V. Three Forgotten Proofs.- A. Perott's Proof.- B. Auric's Proof.- C.…