193,603 research outputs found
Spacetime and Euclidean Geometry
Using only the principle of relativity and Euclidean geometry we show in this
pedagogical article that the square of proper time or length in a
two-dimensional spacetime diagram is proportional to the Euclidean area of the
corresponding causal domain. We use this relation to derive the Minkowski line
element by two geometric proofs of the "spacetime Pythagoras theorem".Comment: 11 pages, 9 figures; for a festschrift honoring Michael P. Ryan; v.2:
References to related work adde
Menelaus's theorem for hyperbolic quadrilaterals in the Einstein relativistic velocity model of hyperbolic geometry
Hyperbolic Geometry appeared in the first half of the 19th century as an attempt to understand Euclid's axiomatic basis of Geometry. It is also known as a type of non-Euclidean Geometry, being in many respects similar to Euclidean Geometry
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
Non-Euclidean geometry in nature
I describe the manifestation of the non-Euclidean geometry in the behavior of
collective observables of some complex physical systems. Specifically, I
consider the formation of equilibrium shapes of plants and statistics of sparse
random graphs. For these systems I discuss the following interlinked questions:
(i) the optimal embedding of plants leaves in the three-dimensional space, (ii)
the spectral statistics of sparse random matrix ensembles.Comment: 52 pages, 21 figures, last section is rewritten, a reference to
chaotic Hamiltonian systems is adde
Partialy Paradoxist Smarandache Geometries
A paradoxist Smarandache geometry combines Euclidean, hyperbolic, and elliptic geometry into one space along with other non-Euclidean behaviors oflines that would seem to require a discrete space. A class of continuous spaces is presented here together with specific examples that emibit almost all of these phenomena and suggest the prospect of a continuous paradoxist geometry
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