8 (eight) is the natural number following 7 and preceding 9.

← 7 8 9 →
−1 0 1 2 3 4 5 6 7 8 9
Cardinaleight
Ordinal8th
(eighth)
Numeral systemoctal
Factorization23
Divisors1, 2, 4, 8
Greek numeralΗ´
Roman numeralVIII, viii
Greek prefixocta-/oct-
Latin prefixocto-/oct-
Binary10002
Ternary223
Senary126
Octal108
Duodecimal812
Hexadecimal816
Greekη (or Η)
Arabic, Kurdish, Persian, Sindhi, Urdu٨
Amharic
Bengali
Chinese numeral八,捌
Devanāgarī
Kannada
Malayalam
Telugu
Tamil
Hebrewח
Khmer
Thai
ArmenianԸ ը

Etymology edit

English eight, from Old English eahta, æhta, Proto-Germanic *ahto is a direct continuation of Proto-Indo-European *oḱtṓ(w)-, and as such cognate with Greek ὀκτώ and Latin octo-, both of which stems are reflected by the English prefix oct(o)-, as in the ordinal adjective octaval or octavary, the distributive adjective is octonary. The adjective octuple (Latin octu-plus) may also be used as a noun, meaning "a set of eight items"; the diminutive octuplet is mostly used to refer to eight siblings delivered in one birth.

The Semitic numeral is based on a root *θmn-, whence Akkadian smn-, Arabic ṯmn-, Hebrew šmn- etc. The Chinese numeral, written (Mandarin: ; Cantonese: baat), is from Old Chinese *priāt-, ultimately from Sino-Tibetan b-r-gyat or b-g-ryat which also yielded Tibetan brgyat.

It has been argued that, as the cardinal number 7 is the highest number of items that can universally be cognitively processed as a single set, the etymology of the numeral eight might be the first to be considered composite, either as "twice four" or as "two short of ten", or similar. The Turkic words for "eight" are from a Proto-Turkic stem *sekiz, which has been suggested as originating as a negation of eki "two", as in "without two fingers" (i.e., "two short of ten; two fingers are not being held up");[1] this same principle is found in Finnic *kakte-ksa, which conveys a meaning of "two before (ten)". The Proto-Indo-European reconstruction *oḱtṓ(w)- itself has been argued as representing an old dual, which would correspond to an original meaning of "twice four". Proponents of this "quaternary hypothesis" adduce the numeral 9, which might be built on the stem new-, meaning "new" (indicating the beginning of a "new set of numerals" after having counted to eight).[2]

Evolution of the Arabic digit edit

 
Evolution of the numeral 8 from the Indians to the Europeans

The modern digit 8, like all modern Arabic numerals other than zero, originates with the Brahmi numerals. The Brahmi digit for eight by the 1st century was written in one stroke as a curve └┐ looking like an uppercase H with the bottom half of the left line and the upper half of the right line removed. However, the digit for eight used in India in the early centuries of the Common Era developed considerable graphic variation, and in some cases took the shape of a single wedge, which was adopted into the Perso-Arabic tradition as ٨ (and also gave rise to the later Devanagari form ); the alternative curved glyph also existed as a variant in Perso-Arabic tradition, where it came to look similar to our digit 5.[year needed]

The digits as used in Al-Andalus by the 10th century were a distinctive western variant of the glyphs used in the Arabic-speaking world, known as ghubār numerals (ghubār translating to "sand table"). In these digits, the line of the 5-like glyph used in Indian manuscripts for eight came to be formed in ghubār as a closed loop, which was the 8-shape that became adopted into European use in the 10th century.[3]

Just as in most modern typefaces, in typefaces with text figures the character for the digit 8 usually has an ascender, as, for example, in  .

The infinity symbol ∞, described as a "sideways figure eight", is unrelated to the digit 8 in origin; it is first used (in the mathematical meaning "infinity") in the 17th century, and it may be derived from the Roman numeral for "one thousand" CIƆ, or alternatively from the final Greek letter, ω.

In mathematics edit

Eight is the third composite number, lying between the fourth prime number (7) and the fourth composite number (9). 8 is the first non-unitary cube prime of the form p3. With proper divisors 1, 2, and 4, it is the third power of two (23). 8 is the first number which is neither prime nor semiprime and the only nonzero perfect power that is one less than another perfect power, by Mihăilescu's Theorem.

Sphenic numbers always have exactly eight divisors.[10]

A polygon with eight sides is an octagon.[11] The sides and span of a regular octagon, or truncated square, are in 1 : 1 + 2 silver ratio, and its circumscribing square has a side and diagonal length ratio of 1 : 2; with both the silver ratio and the square root of two intimately interconnected through Pell numbers, where in particular the quotient of successive Pell numbers generates rational approximations for coordinates of a regular octagon.[12][13] With a central angle of 45 degrees and an internal angle of 135 degrees, a regular octagon can fill a plane-vertex with a regular triangle and a regular icositetragon, as well as tessellate two-dimensional space alongside squares in the truncated square tiling. This tiling is one of eight Archimedean tilings that are semi-regular, or made of more than one type of regular polygon, and the only tiling that can admit a regular octagon.[14] The Ammann–Beenker tiling is a nonperiodic tesselation of prototiles that feature prominent octagonal silver eightfold symmetry, that is the two-dimensional orthographic projection of the four-dimensional 8-8 duoprism.[15] In number theory, figurate numbers representing octagons are called octagonal numbers.[16]

A cube is a regular polyhedron with eight vertices that also forms the cubic honeycomb, the only regular honeycomb in three-dimensional space.[17] Through various truncation operations, the cubic honeycomb generates eight other convex uniform honeycombs under the cubic group  .[18] The octahedron, with eight equilateral triangles as faces, is the dual polyhedron to the cube and one of eight convex deltahedra.[19][20] The stella octangula, or eight-pointed star, is the only stellation with octahedral symmetry. It has eight triangular faces alongside eight vertices that forms a cubic faceting, composed of two self-dual tetrahedra that makes it the simplest of five regular compounds. The cuboctahedron, on the other hand, is a rectified cube or rectified octahedron, and one of only two convex quasiregular polyhedra. It contains eight equilateral triangular faces, whose first stellation is the cube-octahedron compound.[21][22] There are also eight uniform polyhedron compounds made purely of octahedra, including the regular compound of five octahedra, and an infinite amount of polyhedron compounds made only of octahedra as triangular antiprisms (UC22 and UC23, with p = 3 and q = 1).

The truncated tetrahedron is the simplest Archimedean solid, made of four triangles and four hexagons, the hexagonal prism, which classifies as an irregular octahedron and parallelohedron, is able to tessellate space as a three-dimensional analogue of the hexagon, and the gyrobifastigium, with four square faces and four triangular faces, is the only Johnson solid that is able to tessellate space. The truncated octahedron, also a parallelohedron, is the permutohedron of order four, with eight hexagonal faces alongside six squares is likewise the only Archimedean solid that can generate a honeycomb on its own.

A tesseract or 8-cell is the four-dimensional analogue of the cube. It is one of six regular polychora, with a total of eight cubical cells, hence its name. Its dual figure is the analogue of the octahedron, with twice the amount of cells and simply termed the 16-cell, that is the orthonormal basis of vectors in four dimensions. Whereas a tesseractic honeycomb is self-dual, a 16-cell honeycomb is dual to a 24-cell honeycomb that is made of 24-cells. The 24-cell is also regular, and made purely of octahedra whose vertex arrangement represents the ring of Hurwitz integral quaternions. Both the tesseract and the 16-cell can fit inside a 24-cell, and in a 24-cell honeycomb, eight 24-cells meet at a vertex. Also, the Petrie polygon of the tesseract and the 16-cell is a regular octagon.

Vertex-transitive semiregular polytopes whose facets are finite exist up through the 8th dimension. In the third dimension, they include the Archimedean solids and the infinite family of uniform prisms and antiprisms, while in the fourth dimension, only the rectified 5-cell, the rectified 600-cell, and the snub 24-cell are semiregular polytopes. For dimensions five through eight, the demipenteract and the k21 polytopes 221, 321, and 421 are the only semiregular (Gosset) polytopes. Collectively, the k21 family of polytopes contains eight figures that are rooted in the triangular prism, which is the simplest semiregular polytope that is made of three cubes and two equilateral triangles. It also includes one of only three semiregular Euclidean honeycombs: the affine 521 honeycomb that represents the arrangement of vertices of the eight-dimensional   lattice, and made of 421 facets. The culminating figure is the ninth-dimensional 621 honeycomb, which is the only affine semiregular paracompact hyperbolic honeycomb with infinite facets and vertex figures in the k21 family. There are no other finite semiregular polytopes or honeycombs in dimensions n > 8.

The octonions are a hypercomplex normed division algebra that are an extension of the complex numbers. They are realized in eight dimensions, where they have an isotopy group over the real numbers that is spin group Spin(8), the unique such group that exhibits a phenomenon of triality. As a double cover of special orthogonal group SO(8), Spin(8) contains the special orthogonal Lie algebra D4 as its Dynkin diagram, whose order-three outer automorphism is isomorphic to the symmetric group S3, giving rise to its triality. Over finite fields, the eight-dimensional Steinberg group 3D4(q3) is simple, and one of sixteen such groups in the classification of finite simple groups. As is Lie algebra E8, whose complex form in 248 dimensions is the largest of five exceptional Lie algebras that include E7 and E6, which are held inside E8. The smallest such algebra is G2, that is the automorphism group of the octonions. In mathematical physics, special unitary group SO(3) has an eight-dimensional adjoint representation whose colors are ascribed gauge symmetries that represent the vectors of the eight gluons in the Standard Model.

The number 8 is involved with a number of interesting mathematical phenomena related to the notion of Bott periodicity. If   is the direct limit of the inclusions of real orthogonal groups  , the following holds:

 .

Clifford algebras also display a periodicity of 8.[23] For example, the algebra Cl(p + 8,q) is isomorphic to the algebra of 16 by 16 matrices with entries in Cl(p,q). We also see a period of 8 in the K-theory of spheres and in the representation theory of the rotation groups, the latter giving rise to the 8 by 8 spinorial chessboard. All of these properties are closely related to the properties of the octonions, which occupy the highest possible dimension for a normed division algebra.

The   lattice Γ8 is the smallest positive even unimodular lattice. As a lattice, it holds the optimal structure for the densest packing of 240 spheres in eight dimensions, whose lattice points also represent the root system of Lie group E8. This honeycomb arrangement is shared by a unique complex tessellation of Witting polytopes, also with 240 vertices. Each complex Witting polytope is made of Hessian polyhedral cells that have Möbius–Kantor polygons as faces, each with eight vertices and eight complex equilateral triangles as edges, whose Petrie polygons form regular octagons. In general, positive even unimodular lattices only exist in dimensions proportional to eight. In the 16th dimension, there are two such lattices : Γ8 ⊕ Γ8 and Γ16, while in the 24th dimension there are precisely twenty-four such lattices that are called the Niemeier lattices, the most important being the Leech lattice, which can be constructed using the octonions as well as with three copies of the ring of icosians that are isomorphic to the   lattice.[24][25] The order of the smallest non-abelian group all of whose subgroups are normal is 8.

In the classification of sporadic groups, the third generation consists of eight groups, four of which are centralizers of   (itself the largest group of this generation), with another three transpositions of Fischer group  .[26] 8 is the difference between 53 and 61, which are the two smallest prime numbers that do not divide the order of any sporadic group. The largest supersingular prime that divides the order of   is 71, which is the eighth self-convolution of Fibonacci numbers (where 744, which is essential to Moonshine theory, is the twelfth).[27][28] While only two sporadic groups have eight prime factors in their order (Lyons group   and Fischer group  ), Mathieu group   holds a semi-presentation whose order is equal to  .[29]

List of basic calculations edit

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
8 × x 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
8 ÷ x 8 4 2.6 2 1.6 1.3 1.142857 1 0.8 0.8 0.72 0.6 0.615384 0.571428 0.53
x ÷ 8 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 1.125 1.25 1.375 1.5 1.625 1.75 1.875
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
8x 8 64 512 4096 32768 262144 2097152 16777216 134217728 1073741824 8589934592 68719476736 549755813888
x8 1 256 6561 65536 390625 1679616 5764801 16777216 43046721 100000000 214358881 429981696 815730721

In other bases edit

A number is divisible by 8 if its last three digits, when written in decimal, are also divisible by 8, or its last three digits are 0 when written in binary.

8 is the base of the octal number system, which is mostly used with computers.[30] In octal, one digit represents three bits. In modern computers, a byte is a grouping of eight bits, also called an octet.

In science edit

Physics edit

Astronomy edit

Chemistry edit

Geology edit

  • A disphenoid crystal is bounded by eight scalene triangles arranged in pairs. A ditetragonal prism in the tetragonal crystal system has eight similar faces whose alternate interfacial angles only are equal.

Biology edit

  • All spiders, and more generally all arachnids, have eight legs.[38] Orb-weaver spiders of the cosmopolitan family Areneidae have eight similar eyes.[39]
  • The octopus and its cephalopod relatives in genus Argonauta have eight arms (tentacles).
  • Compound coelenterates of the subclass or order Alcyonaria have polyps with eight-branched tentacles and eight septa.[40]
  • Sea anemones of genus Edwardsia have eight mesenteries.[41]
  • Animals of phylum Ctenophora swim by means of eight meridional bands of transverse ciliated plates, each plate representing a row of large modified cilia.[42]
  • The eight-spotted forester (genus Alypia, family Zygaenidae) is a diurnal moth having black wings with brilliant white spots.
  • The ascus in fungi of the class Ascomycetes, following nuclear fusion, bears within it typically eight ascospores.[43]
  • Herbs of genus Coreopsis (tickseed) have showy flower heads with involucral bracts in two distinct series of eight each.
  • In human adult dentition there are eight teeth in each quadrant.[44] The eighth tooth is the so-called wisdom tooth.
  • There are eight cervical nerves on each side in man and most mammals.[45]

Psychology edit

In technology edit

 
NATO signal flag for 8

In measurement edit

In culture edit

Currency edit

  • Sailors and civilians alike from the 1500s onward referred to evenly divided parts of the Spanish dollar as "pieces of eight", or "bits".

Architecture edit

  • Various types of buildings are usually eight-sided (octagonal), such as single-roomed gazebos[56] and multi-roomed pagodas (descended from stupas; see religion section below).
  • Eight caulicoles rise out of the leafage in a Corinthian capital, ending in leaves that support the volutes.

In religion, folk belief and divination edit

Hinduism edit

  • As sourced from the Mahabharata, there are 8 vasus who are given elemental names:[57]
    1. Anala or Agni (fire)
    2. Dhara or Prithvi (earth)
    3. Anila or Vayudeva (wind)
    4. Apa (water)
    5. Prabhasa or Dyauh (sky)
    6. Pratyusha
    7. Soma
    8. Dhruva
  • The goddess of wealth and prosperity, Lakshmi, has eight forms known as Ashta Lakshmi and worshipped as:
    "Maha-lakshmi, Dhana-lakshmi, Dhanya-lakshmi, Gaja-lakshmi,
    Santana-lakshmi, Veera-lakshmi, Vijaya-lakshmi and Vidhya-lakshmi
    "[58]
  • There are eight nidhi, or seats of wealth, according to Hinduism.
  • There are eight guardians of the directions known as Astha-dikpalas.[59]
  • There are eight Hindu monasteries established by the saint Madhvacharya in Udupi, India popularly known as the Ashta Mathas of Udupi.[60]

Buddhism edit

 
In Buddhism, the 8-spoked Dharmacakra represents the Noble Eightfold Path.
  • The Dharmacakra, a Buddhist symbol, has eight spokes.[61] The Buddha's principal teaching—the Four Noble Truths—ramifies as the Noble Eightfold Path and the Buddha emphasizes the importance of the eight attainments or jhanas.
  • In Mahayana Buddhism, the branches of the Eightfold Path are embodied by the Eight Great Bodhisattvas: (Manjusri, Vajrapani, Avalokiteśvara, Maitreya, Ksitigarbha, Nivaranavishkambhi, Akasagarbha, and Samantabhadra).[62] These are later (controversially) associated with the Eight Consciousnesses according to the Yogacara school of thought: consciousness in the five senses, thought-consciousness, self-consciousness, and unconsciousness-"consciousness" or "store-house consciousness" (alaya-vijñana). The "irreversible" state of enlightenment, at which point a Bodhisattva goes on "autopilot", is the Eight Ground or bhūmi. In general, "eight" seems to be an auspicious number for Buddhists, e.g., the "eight auspicious symbols" (the jewel-encrusted parasol; the goldfish (always shown as a pair, e.g., the glyph of Pisces); the self-replenishing amphora; the white kamala lotus-flower; the white conch; the eternal (Celtic-style, infinitely looping) knot; the banner of imperial victory; the eight-spoked wheel that guides the ship of state, or that symbolizes the Buddha's teaching). Similarly, Buddha's birthday falls on the 8th day of the 4th month of the Chinese calendar.

Judaism edit

Christianity edit

Islam edit

 
The octagram Rub el Hizb

Taoism edit

Other edit

As a lucky number edit

  • The number eight is considered to be a lucky number in Chinese and other Asian cultures.[77] Eight (; accounting ; pinyin ) is considered a lucky number in Chinese culture because it sounds like the word meaning to generate wealth (發(T) 发(S); Pinyin: ). Property with the number 8 may be valued greatly by Chinese. For example, a Hong Kong number plate with the number 8 was sold for $640,000.[78] The opening ceremony of the Summer Olympics in Beijing started at 8 seconds and 8 minutes past 8 pm (local time) on 8 August 2008.[79]
  • In Pythagorean numerology (a pseudoscience) the number 8 represents victory, prosperity and overcoming.
  • Eight (, hachi, ya) is also considered a lucky number in Japan, but the reason is different from that in Chinese culture.[80] Eight gives an idea of growing prosperous, because the letter () broadens gradually.
  • The Japanese thought of eight (, ya) as a holy number in the ancient times. The reason is less well-understood, but it is thought that it is related to the fact they used eight to express large numbers vaguely such as manyfold (やえはたえ, Yae Hatae) (literally, eightfold and twentyfold), many clouds (やくも, Yakumo) (literally, eight clouds), millions and millions of Gods (やおよろずのかみ, Yaoyorozu no Kami) (literally, eight millions of Gods), etc. It is also guessed that the ancient Japanese gave importance to pairs, so some researchers guess twice as four (, yo), which is also guessed to be a holy number in those times because it indicates the world (north, south, east, and west) might be considered a very holy number.
  • In numerology, 8 is the number of building, and in some theories, also the number of destruction.

In astrology edit

In music and dance edit

In film and television edit

In sports and other games edit

 
An 8-ball in pool
  • Eight-ball pool is played with a cue ball and 15 numbered balls, the black ball numbered 8 being the middle and most important one, as the winner is the player or side that legally pockets it after first pocketing its numerical group of 7 object balls (for other meanings see Eight ball (disambiguation)).
  • In chess, each side has eight pawns and the board is made of 64 squares arranged in an eight by eight lattice. The eight queens puzzle is a challenge to arrange eight queens on the board so that none can capture any of the others.
  • In the game of eights or Crazy Eights, each successive player must play a card either of the same suit or of the same rank as that played by the preceding player, or may play an eight and call for any suit. The object is to get rid of all one's cards first.
  • In association football, the number 8 has historically been the number of the Central Midfielder.
  • In Australian rules football, the top eight teams at the end of the Australian Football League regular season qualify for the finals series (i.e. playoffs).
  • In baseball:
  • In rugby union, the only position without a proper name is the Number 8, a forward position.
  • In rugby league:
    • Most competitions (though not the Super League, which uses static squad numbering) use a position-based player numbering system in which one of the two starting props wears the number 8.
    • The Australia-based National Rugby League has its own 8-team finals series, similar but not identical in structure to that of the Australian Football League.
  • In rowing, an "eight" refers to a sweep-oar racing boat with a crew of eight rowers plus a coxswain.[108]
  • In the 2008 Games of the XXIX Olympiad held in Beijing, the official opening was on 08/08/08 at 8:08:08 p.m. CST.
  • In rock climbing, climbers frequently use the figure-eight knot to tie into their harnesses.
  • The Women's College World Series, the final phase of the NCAA Division I softball tournament, like its men's counterpart in baseball, features eight teams.
  • In curling an 8-point 'Eight Ender' is a perfect end. Each team delivers 8 Stones per end.

In foods edit

  • Nestlé sells a brand of chocolates filled with peppermint-flavoured cream called After Eight, referring to the time 8 p.m.[109]
  • There are eight vegetables in V8 juice.[110]

In literature edit

  • Eights may refer to octosyllabic, usually iambic, lines of verse.
  • The drott-kvaett, an Old Icelandic verse, consisted of a stanza of eight regular lines.[111]
  • In Terry Pratchett's Discworld series, eight is a magical number[112] and is considered taboo. Eight is not safe to be said by wizards on the Discworld and is the number of Bel-Shamharoth. Also, there are eight days in a Disc week and eight colours in a Disc spectrum, the eighth one being octarine.
  • Lewis Carroll's poem The Hunting of the Snark has 8 "fits" (cantos), which is noted in the full name "The Hunting of the Snark – An Agony, in Eight Fits".[113]
  • Eight apparitions appear to Macbeth in Act 4 scene 1 of Shakespeare's Macbeth as representations of the eight descendants of Banquo.

In slang edit

Other uses edit

  • A figure 8 is the common name of a geometric shape, often used in the context of sports, such as skating.[116] Figure-eight turns of a rope or cable around a cleat, pin, or bitt are used to belay something.[117]

See also edit

References edit

  1. ^ Etymological Dictionary of Turkic Languages: Common Turkic and Interturkic stems starting with letters «L», «M», «N», «P», «S», Vostochnaja Literatura RAS, 2003, 241f. (altaica.ru Archived 31 October 2007 at the Wayback Machine)
  2. ^ the hypothesis is discussed critically (and rejected as "without sufficient support") by Werner Winter, 'Some thought about Indo-European numerals' in: Jadranka Gvozdanović (ed.), Indo-European Numerals, Walter de Gruyter, 1992, 14f.
  3. ^ Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 395, Fig. 24.68.
  4. ^ Sloane, N. J. A. (ed.). "Sequence A076980 (Leyland numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  5. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 88
  6. ^ Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 8 August 2023.
  7. ^ Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbersnumbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 24 September 2023.
  8. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers: a(n) is the binomial(n+1,2) equal to n*(n+1)/2 and 0 + 1 + 2 + ... + n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 24 September 2023.
  9. ^ Sloane, N. J. A. (ed.). "Sequence A054377 (Primary pseudoperfect numbers: numbers n > 1 such that 1/n + sum 1/p is 1, where the sum is over the primes p | n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 January 2024.
  10. ^ Weisstein, Eric W. "Sphenic Number". mathworld.wolfram.com. Retrieved 7 August 2020. ...then every sphenic number n=pqr has precisely eight positive divisors
  11. ^ Weisstein, Eric W. "Octagon". mathworld.wolfram.com. Retrieved 7 August 2020.
  12. ^ Bicknell, Marjorie (1975). "A primer on the Pell sequence and related sequences". Fibonacci Quarterly. 13 (4): 345–349. MR 0387173.
  13. ^ Knuth, Donald E. (1994). "Leaper graphs". The Mathematical Gazette. 78 (483): 283. arXiv:math.CO/9411240. Bibcode:1994math.....11240K. doi:10.2307/3620202. JSTOR 3620202. S2CID 16856513.
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  19. ^ Freudenthal, H; van der Waerden, B. L. (1947), "Over een bewering van Euclides ("On an Assertion of Euclid")", Simon Stevin (in Dutch), 25: 115–128
  20. ^ Roger Kaufman. "The Convex Deltahedra And the Allowance of Coplanar Faces". The Kaufman Website. Retrieved 25 June 2022.
  21. ^ Weisstein, Eric W. "Cuboctahedron". mathworld.wolfram.com. Retrieved 25 June 2022.
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  29. ^ Wilson, R.A.; Parker, R.A.; et al. (1999). "ATLAS: Mathieu group M23". ATLAS of Finite Group Representations.
  30. ^ Weisstein, Eric W. "Octal". mathworld.wolfram.com. Retrieved 7 August 2020.
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