steradian

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International System of Units
International System of Units

(geometry) In the International System of Units, the derived unit of solid angle; the solid angle subtended at the centre of a sphere of radius r by a portion of the surface of the sphere that has area r². Symbol: sr


equator

From Late Latin (circulus) aequator (diei et noctis).

bor

(astronomy) A similar great circle on any sphere, especially on a celestial body, or on other reasonably symmetrical three-dimensional body.


spheroid

en-adj

spheroid (comparative more spheroid, superlative most spheroid)

more spheroid (comparative)
most spheroid (superlative)

Of a shape similar to a squashed sphere.


great circle route

en-noun

great circle route (plural great circle routes)

A route, taken by intercontinental aircraft for instance, which is the shortest distance between two destinations on the surface of the Earth; an arc of a great circle.


great circle

(geometry) A circle defined as the intersection of the surface of a sphere and a plane which passes through the centre of the sphere.

Wherever a navigator may be, it certainly would be to his advantage to be able to find the great circle, or shortest course, and the distance, to his port of destination, [....]

1902, S. L. Penfield, “Use of the Stereographic Projection for Geographical Maps and Sailing Charts”, in American Journal of Science:

The shortest distance between two points on an ocean Mercator chart is not a straight line - it is a curve known as a 'great circle'.

2003, Jeff Toghill, The Navigator's Handbook: Essential Methods and Equipment--and how to Use Them:

Because the Earth's surface is curved, the shortest distance is actually a curve over the surface called a great circle.

2005, Phil Medina, Homework Helpers: Earth Science, page 52:

A segment of such circle representing the shortest distance between two points on the surface of a sphere.


antipodes

From Latin, from Ancient Greek ἀντίποδες (antípodes), from ἀντί (antí, “opposite”) + πόδες (pódes), plural of πούς (poús, “foot”).

[…]a delighted shout from the children swung him toward the door again. His sister, Mrs. Gerard, stood there in carriage gown and sables, radiant with surprise. ¶ "Phil! You! Exactly like you, Philip, to come strolling in from the antipodes—dear fellow!" recovering from the fraternal embrace and holding both lapels of his coat in her gloved hands.

1907 August, Robert W[illiam] Chambers, chapter I, in The Younger Set, New York, N.Y.: D. Appleton & Company, →OCLC:


solid angle

(geometry) The three-dimensional analog of an angle, equivalent to the area of that portion of the surface of a unit sphere that it subtends.


geodesic

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(spherical geometry) A segment of a great circle.


firmament

firmament (countable and uncountable, plural firmaments)

From Middle English firmament, furmament (“heaven; sky”), from Old French firmament (“firmament”), or from its etymon Latin firmāmentum (“support; sky”), from
firmāre (“to strengthen”) + -mentum (suffix indicating an instrument or medium, or the result of an action). Firmāre is the present active infinitive of firmō (“to make firm, strengthen”), from firmus (“firm, strong, stable”), ultimately from Proto-Indo-European *dʰer- (“to hold; to support”).
The Latin word was used in the Vulgate version of the Bible to translate the
Ancient Greek στερέωμα (steréōma, “foundation, framework; firmament”) in the Septuagint (Greek Old Testament), which in turn was used to translate the Hebrew רָקִיעַ (rāqī́aʿ, “celestial dome, vault of heaven”), from the root ר־ק־ע (r-q-`); in Classical Syriac the similar root ܪ-ܩ-ܥ (r-q-ʿ) (related to compacting) gave rise to ܪܩܝܥܐ (rəqīʿā, “compact; firm; firmament, heavens, sky; celestial sphere”).

ref

From Petrus Apianus, Gemma Frisius (1539) Petri Apiani cosmographia, per Gemmam Phrysium, apud Louanienses medicum ac mathematicum insignem, restituta: additis de adem re ipsius Gemmae Phry. libellis, ut sequens pagina docet, Antwerp: In pingui gallina Arnoldo Berckma[n]no [Arnold Birckmann], →OCLC.

ሰማይ

vault of the heavens — see also heavens, sky

ala

field or sphere of an activity or interest — see also field, sphere

[B]etweene the ſphere of Saturne and the Firmament, there is ſuch an incredible and vaſt ſpace or diſtance (7000000. ſemidiameters of the earth, as Tycho [Brahe] calculates) void of ſtarres: [...]

1638, Democritus Junior [pseudonym; Robert Burton], “Ayre Rectified. With a Digression of the Ayre.”, in The Anatomy of Melancholy. […], 5th edition, Oxford, Oxfordshire: […] [Robert Young, Miles Flesher, and Leonard Lichfield and William Turner] for Henry Cripps, →OCLC, partition 2, section 2, member 3, page 254:

But if we yet rise higher, and consider the fixed stars as so many vast oceans of flame, that are each of them attended with a different set of planets, and still discover new firmaments and new lights, that are sunk farther in those unfathomable depths of ether, so as not to be seen by the strongest of our telescopes, we are lost in such a labyrinth of suns and worlds, and confounded with the immensity and magnificence of nature.

1712 July 13 (Gregorian calendar), [Joseph Addison], “WEDNESDAY, July 2, 1712. Paper X. On the Pleasures of the Imagination.”, in The Spectator, number 420; republished in Alexander Chalmers, editor, The Spectator; a New Edition, […], volume IV, New York, N.Y.: D[aniel] Appleton & Company, 1853, →OCLC, page 79:

The World Cœlestial. Containing 11 Sphæres or Heavens, ſay the Theologians and Aſtronomers. [...] Theſe are called the Primum Mobile, the Chriſtaline Heaven, the Firmament adorned with the Fixed Stars, and the Heavens of the Seven Planets.

1715, H[enry] Curson, “A Cosmographical Introduction to the Geographical Description of the World”, in A New Description of the World. […], 2nd edition, […] Benj[amin] Barker, […], →OCLC:

(uncountable, astronomy, historical) In the geocentric Ptolemaic system, the eighth celestial sphere which carried the fixed stars; (countable, by extension) any celestial sphere.


hypersphere

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To prove that the hypersphere is a manifold we wish to divide it into polyhedral solids.

2014, Donald W. Blackett, Elementary Topology: A Combinatorial and Algebraic Approach, Academic Press, →ISBN, page 198:

(geometry) The set of all points in a given hyperspace that are at a given distance from a given point; a generalization of a sphere in four or more dimensions.

гиперсфера

the points of a hyperspace at the same distance from a given point


spherical

en-adj

spherical (comparative more spherical, superlative most spherical)

more spherical (comparative)
most spherical (superlative)

English spheric [from Latin sphaericus, from Ancient Greek σφαιρικός (sphairikós)] + -al.

The ability of a segment of a glass sphere to magnify whatever is placed before it was known around the year 1000, when the spherical segment was called a reading stone, essentially what today we might term a frameless magnifying glass or plain glass paperweight.

2013 September-October, Henry Petroski, “The Evolution of Eyeglasses”, in American Scientist:


celestial sphere

en-noun

celestial sphere (plural celestial spheres)

(astronomy) The apparent outer limit of space; an abstract sphere of infinite radius which serves as the imaginary backdrop for celestial objects, and of which the visible sky is one hemisphere.


small circle

a circle formed by a plane that cuts a sphere not through its centre


spherical wedge

spherical wedge (plural spherical wedges)

(geometry) A volume of a sphere lying between two vertical planes through the centre of the sphere.


spherical lune

(geometry) A portion of the surface of a sphere cut off by two vertical planes through the centre of the sphere.


spherical geometry

spherical geometry (usually uncountable, plural spherical geometries)

(geometry, uncountable) The geometry of the 2-dimensional surface of a sphere;

The geometry of the 2-dimensional surface of a sphere;
(countable) a given geometry of the surface of a sphere; a geometry of the surface of a given sphere, regarded as distinct from that of other spheres.


antipodal

On opposite sides of the globe; pertaining to antipodes


spheroidal

en-adj

spheroidal (comparative more spheroidal, superlative most spheroidal)

more spheroidal (comparative)
most spheroidal (superlative)

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The buccal mass in milacids is spheroidal, and contains an oxygnathic jaw.

2004, G. M. Barker, Natural Enemies of Terrestrial Molluscs, page 299:


celestial equator

en-noun

celestial equator (plural celestial equators)

(astronomy) A great circle on the celestial sphere, coincident with the plane of Earth's Equator.


henagon

From Ancient Greek: εἷς (heîs, “one”, accusative héna) + -gon.

A polygon having one edge and one angle.


monogon

According to Theorem 4.1.1, such a derived imbedding could be obtained from an imbedded voltage graph with one vertex, 6s#x2B;2 edges, and 4s#x2B;2 faces. Of these faces, 4s#x2B;1 should be 3-sided and satisfy KVL. The other face should be a monogon whose net voltage has order two.

1987, Jonathan L. Gross, Thomas W. Tucker, Topological Graph Theory, Dover Publications, published 2001, →ISBN, page 231:

2002, Tao Li, "Laminar Branched Surfaces in 3–manifolds", Geometry & Topology 6, page 158,
There is no monogon in M-int(N(B)), ie, no disk D⊂M-int(N(B)) with ∂D=D∩N(B)=α∪β, where α⊂∂_(vN)(B) is in an interval fiber of ∂_(vN)(B) and β⊂∂_(hN)(B).

a. 2006, Thilo Kuessner, "A survey on simplicial volume and invariants of foliations and laminations", in, Paweł Walczak, et al., editors, Foliations 2005, →ISBN, page 295,
An end-compressing monogon for F is a monogon properly embedded in the complimentary region C which is not homotopic (rel. boundary) into ∂C.

(geometry) A two-dimensional object comprising one vertex, one edge both of whose ends are that vertex, and one face filling in the hollow formed by that edge.


digon

From di- (prefix meaning ‘two’) + -gon (suffix forming the names of plane figures containing a given number of angles).

confix

monogon · digon · trigon · tetragon · pentagon · hexagon · heptagon · octagon · nonagon · decagon · undecagon · dodecagon · see also Category:English terms suffixed with -gon

They [the students] also came upon new and unusual mathematical figures: the digon, a two-sided polygon on a spherical space, and the apeirogon, an open polygon with infinitely many sides […]. All these discoveries brought up even more questions. Is a circle a polygon? What makes an octagon an octagon – its eight vertices, its eight sides, or both? Can a polygon cross itself? Does a polygon need to be closed?

2013, Brent Davis, Moshe Renert, chapter 6, in The Math Teachers Know: Profound Understanding of Emergent Mathematics, New York, N.Y., Abingdon, Oxfordshire: Routledge, →ISBN, page 102:

(geometry) A polygon having two edges and two vertices.

digono

A pair of parallel undirected edges in a multigraph.

A pair of antiparallel edges in a directed graph.


horoball

Coined by William Thurston (1946-2012), American mathematician. See horosphere and ball.

(hyperbolic geometry) A horosphere together with the space above it.


horosphere

A modern scientific coinage, probably from a Latinized spelling of Ancient Greek ὅρος (hóros, “boundary”) (not to be confused with ὧρος (hôros, “year”), which gave rise to many time-related terms) + sphere

An n-1-dimensional hyperplane in hyperbolic n-dimensional space: it is (in the Poincaré disk) Euclidean-tangent at infinity to the boundary sphere or (in the upper-half-space model) Euclidean-parallel to the boundary hyperplane.


bigon

(rare) A polygon having two edges and two vertices.


exotic sphere

A manifold (of any dimension) homeomorphic to a sphere but not diffeomorphic to the standard sphere.


spherium

(physics, mathematics) A model of a form of atom composed of two electrons restricted to the surface of a sphere.


hosohedron

A tessellation of lunes on a spherical surface, such that each lune shares the same two vertices.


hyperball

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We find packing densities using congruent hyperballs and determine the smallest density upper bound of non-congruent hyperball packings generated by the above tilings..

2015, Jenő Szirmai, “Density upper bound for congruent and non-congruent hyperball packings generated by truncated regular simplex tilings”, in arXiv:

(geometry) The set of all points that make a hypersphere.


diangle

/ˈdaɪˌæŋɡəl/

The diangle ABCD indicates the entire duration of life, A and C being its commencement and its termination.

1853, “The Science of Medicine”, in The Journal of Psychological Medicine and Mental Pathology, page 241:

Now the wedge 1,2 is mapped into the exterior of the circular diangle C,S₁,1,O,2,S₂,C; both regions are indicated by a shading of the boundary. We now seek the images of the reflected wedges.

1949, Arnold Sommerfeld, translated by Ernst G. Strauss, Partial Differential Equations in Physics, volume 1, Academic Press, page 319:

Let B be a region of F bounded by a closed polygon comprised of segments of geodesics of surface F, and let hₙ be a sequence of nondegenerate diangles of region B whose sides are geodesics of region B and which converge to diangle hₙ.

1969, Siberian Mathematical Journal, Kluwer/Plenum Publishers, page 107:


biangle

/ˈbaɪˌæŋɡəl/

If n planes be drawn through any axis of the right-vector, each of which makes angles π/n with the planes on either side of it, the whole of space is divided into n congruent figures which may be called biangles, the space between any two adjacent planes being easily seen to be continuous with the vertically opposite space between them.

1895, W. Burnside, “Kinematics of Non-Euclidean Space”, in Proceedings of the Edinburgh Mathematical Society, page 46:

Then the right biangle CABD and the oblique biangle CPND are equivalent since the triangles API and BNI are congruent.

1912, Proceedings of the Edinburgh Mathematical Society, volumes 30-32, Scottish Academic Press, page 35:

It is therefore natural to consider whether precisely this concept of a straight line, with which Kant¹¹ too was familiar, is the reason why Kant in the one passage says that the concept of a straight biangle is free of contradiction.

2012 December 6, Gerold Prauss, “Kant and the Straight Biangle”, in Enno Rudolph, Ion-Olimpiu Stamatescu, editors, Philosophy, Mathematics and Modern Physics: A Dialogue, page 228:

non-Euclidean geometry
non-Euclidean geometry#English

A digon or bigon; a two-sided shape (especially in non-Euclidean geometry)


Poincaré ball

The equivalent of a Poincaré disk in three or more dimensions.


spherical coordinates

en-noun

spherical coordinates pl (normally plural, singular spherical coordinate)

A coordinate system, for three-dimensional space, using latitude, longitude and radius (i.e. distance from the center) as the coordinates.


square degree

(geometry) A unit of solid angle equal to π²/32400 steradian


n-sphere

An n-dimensional hypersphere: the 2-sphere is a circle, the 3-sphere is the usual sphere, the 4-sphere is a glome, etc.


goat grazing problem

goat grazing problem (plural goat grazing problems)

The same problem generalized to any number of dimensions: given a unit n-sphere (n-dimensional hypersphere) S, and a point P on the edge of S, what is r such that exactly one-half of the space (content) within S, is within distance r of P?


goat problem

goat grazing problem
goat grazing problem#English