7 Explained

Number:	7
Numeral:	septenary
Factorization:	prime
Prime:	4th
Divisor:	1, 7
Roman:	VII, vii
Greek Prefix:	hepta-/hept-
Latin Prefix:	septua-
Lang1:	Greek numeral
Lang1 Symbol:	Z, ζ
Lang2:	Amharic
Lang2 Symbol:	፯
Lang3:	Arabic, Kurdish, Persian
Lang4:	Sindhi, Urdu
Lang5:	Bengali
Lang6:	Chinese numeral
Lang6 Symbol:	七, 柒
Lang7:	Devanāgarī
Lang8:	Telugu
Lang9:	Tamil
Lang10:	Hebrew
Lang11:	Khmer
Lang11 Symbol:	៧
Lang12:	Thai
Lang12 Symbol:	๗
Lang13:	Kannada
Lang14:	Malayalam
Lang14 Symbol:	൭
Lang15:	Armenian
Lang15 Symbol:	Է
Lang16:	Babylonian numeral
Lang17:	Egyptian hieroglyph
Lang19:	Morse code

7 (seven) is the natural number following 6 and preceding 8. It is the only prime number preceding a cube.

As an early prime number in the series of positive integers, the number seven has greatly symbolic associations in religion, mythology, superstition and philosophy. The seven classical planets resulted in seven being the number of days in a week.^[1] 7 is often considered lucky in Western culture and is often seen as highly symbolic. Unlike Western culture, in Vietnamese culture, the number seven is sometimes considered unlucky.

Evolution of the Arabic digit

For early Brahmi numerals, 7 was written more or less in one stroke as a curve that looks like an uppercase (J) vertically inverted (ᒉ). The western Arab peoples' main contribution was to make the longer line diagonal rather than straight, though they showed some tendencies to making the digit more rectilinear. The eastern Arab peoples developed the digit from a form that looked something like 6 to one that looked like an uppercase V. Both modern Arab forms influenced the European form, a two-stroke form consisting of a horizontal upper stroke joined at its right to a stroke going down to the bottom left corner, a line that is slightly curved in some font variants. As is the case with the European digit, the Cham and Khmer digit for 7 also evolved to look like their digit 1, though in a different way, so they were also concerned with making their 7 more different. For the Khmer this often involved adding a horizontal line to the top of the digit.^[2] This is analogous to the horizontal stroke through the middle that is sometimes used in handwriting in the Western world but which is almost never used in computer fonts. This horizontal stroke is, however, important to distinguish the glyph for seven from the glyph for one in writing that uses a long upstroke in the glyph for 1. In some Greek dialects of the early 12th century the longer line diagonal was drawn in a rather semicircular transverse line. On seven-segment displays, 7 is the digit with the most common graphic variation (1, 6 and 9 also have variant glyphs). Most calculators use three line segments, but on Sharp, Casio, and a few other brands of calculators, 7 is written with four line segments because in Japan, Korea and Taiwan 7 is written with a "hook" on the left, as ① in the following illustration.While the shape of the character for the digit 7 has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender (⁊), as, for example, in .

Most people in Continental Europe,^[3] Indonesia, and some in Britain, Ireland, and Canada, as well as Latin America, write 7 with a line through the middle, sometimes with the top line crooked. The line through the middle is useful to clearly differentiate the digit from the digit one, as the two can appear similar when written in certain styles of handwriting. This form is used in official handwriting rules for primary school in Russia, Ukraine, Bulgaria, Poland, other Slavic countries,^[4] France,^[5] Italy, Belgium, the Netherlands, Finland,^[6] Romania, Germany, Greece,^[7] and Hungary.

In mathematics

Seven, the fourth prime number, is not only a Mersenne prime (since

2 ^{3} - 1 = 7 ) but also a double Mersenne prime since the exponent, 3, is itself a Mersenne prime.^[8] It is also a Newman–Shanks–Williams prime,^[9] a Woodall prime,^[10] a factorial prime,^[11] a Harshad number, a lucky prime,^[12] a happy number (happy prime),^[13] a safe prime (the only), a Leyland prime of the second kind and the fourth Heegner number.^[14] Seven is the lowest natural number that cannot be represented as the sum of the squares of three integers.

A seven-sided shape is a heptagon.^[15] The regular n-gons for n ⩽ 6 can be constructed by compass and straightedge alone, which makes the heptagon the first regular polygon that cannot be directly constructed with these simple tools.^[16] Figurate numbers representing heptagons are called heptagonal numbers.^[17] 7 is also a centered hexagonal number.^[18]

7 is the only number D for which the equation has more than two solutions for n and x natural. In particular, the equation is known as the Ramanujan–Nagell equation. 7 is one of seven numbers in the positive definite quadratic integer matrix representative of all odd numbers: .^{[19] [20]}

There are 7 frieze groups in two dimensions, consisting of symmetries of the plane whose group of translations is isomorphic to the group of integers.^[21] These are related to the 17 wallpaper groups whose transformations and isometries repeat two-dimensional patterns in the plane.^{[22] [23]}

A heptagon in Euclidean space is unable to generate uniform tilings alongside other polygons, like the regular pentagon. However, it is one of fourteen polygons that can fill a plane-vertex tiling, in its case only alongside a regular triangle and a 42-sided polygon (3.7.42).^{[24] [25]} This is also one of twenty-one such configurations from seventeen combinations of polygons, that features the largest and smallest polygons possible.^{[26] [27]} Otherwise, for any regular n-sided polygon, the maximum number of intersecting diagonals (other than through its center) is at most 7.^[28]

In Wythoff's kaleidoscopic constructions, seven distinct generator points that lie on mirror edges of a three-sided Schwarz triangle are used to create most uniform tilings and polyhedra; an eighth point lying on all three mirrors is technically degenerate, reserved to represent snub forms only.^[29]

Seven of eight semiregular tilings are Wythoffian (the only exception is the elongated triangular tiling), where there exist three tilings that are regular, all of which are Wythoffian.^[30] Seven of nine uniform colorings of the square tiling are also Wythoffian.^[31] In two dimensions, there are precisely seven 7-uniform Krotenheerdt tilings, with no other such k-uniform tilings for k > 7, and it is also the only k for which the count of Krotenheerdt tilings agrees with k.^{[32] [33]}

The Fano plane, the smallest possible finite projective plane, has 7 points and 7 lines arranged such that every line contains 3 points and 3 lines cross every point.^[34] This is related to other appearances of the number seven in relation to exceptional objects, like the fact that the octonions contain seven distinct square roots of -1, seven-dimensional vectors have a cross product, and the number of equiangular lines possible in seven-dimensional space is anomalously large.^{[35] [36] [37]}

The lowest known dimension for an exotic sphere is the seventh dimension, with a total of 28 differentiable structures; there may exist exotic smooth structures on the four-dimensional sphere.^{[38] [39]}

In hyperbolic space, 7 is the highest dimension for non-simplex hypercompact Vinberg polytopes of rank n + 4 mirrors, where there is one unique figure with eleven facets.^[40] On the other hand, such figures with rank n + 3 mirrors exist in dimensions 4, 5, 6 and 8; not in 7.

There are seven fundamental types of catastrophes.^[41]

When rolling two standard six-sided dice, seven has a 6 in 6 (or) probability of being rolled (1–6, 6–1, 2–5, 5–2, 3–4, or 4–3), the greatest of any number.^[42] The opposite sides of a standard six-sided die always add to 7.

The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000.^[43] Currently, six of the problems remain unsolved.^[44]

Basic calculations

Division	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
7 ÷ x	7	3.5	2.	1.75	1.4	1.1	1	0.875	0.	0.7	0.	0.58	0.	0.5	0.4
x ÷ 7	0.142857	0.285714	0.428571	0.571428	0.714285	0.857142		1.142857	1.285714	1.428571	1.571428	1.714285	1.857142		2.142857

Exponentiation	1	2	3	4	5	6	7	8	9	10	11	12	13
7^x	7			2401	16807	117649	823543	5764801	40353607	282475249	1977326743	13841287201	96889010407
x^7	1		2187	16384	78125	279936	823543	2097152	4782969		19487171	35831808	62748517

In decimal

In decimal representation, the reciprocal of 7 repeats six digits (as 0.),^{[45] [46]} whose sum when cycling back to 1 is equal to 28.

divided by 7 is exactly . Therefore, when a vulgar fraction with 7 in the denominator is converted to a decimal expansion, the result has the same six-digit repeating sequence after the decimal point, but the sequence can start with any of those six digits.^[47] For example, and

In fact, if one sorts the digits in the number 142,857 in ascending order, 124578, it is possible to know from which of the digits the decimal part of the number is going to begin with. The remainder of dividing any number by 7 will give the position in the sequence 124578 that the decimal part of the resulting number will start. For example, 628 ÷ 7 = ; here 5 is the remainder, and would correspond to number 7 in the ranking of the ascending sequence. So in this case, . Another example,, hence the remainder is 2, and this corresponds to number 2 in the sequence. In this case, .

In science

• Seven colors in a rainbow
• Seven continents
• Seven climes
• The neutral pH balance
• Number of notes in the diatonic scale of Western music
• Number of spots most commonly found on ladybugs
• Atomic number for nitrogen
• Number of diatomic molecules
• Seven basic crystal systems

In psychology

• Seven, plus or minus two as a model of working memory
• Seven psychological types called the Seven Rays in the teachings of Alice A. Bailey
• In Western culture, seven is consistently listed as people's favorite number^{[48] [49]}
• When guessing numbers 1–10, the number 7 is most likely to be picked^[50]
• Seven-year itch, a term that suggests that happiness in a marriage declines after around seven years

Classical antiquity

The Pythagoreans invested particular numbers with unique spiritual properties. The number seven was considered to be particularly interesting because it consisted of the union of the physical (number 4) with the spiritual (number 3).^[51] In Pythagorean numerology the number 7 means spirituality.

References from classical antiquity to the number seven include:

• Seven Classical planets and the derivative Seven Heavens
• Seven Wonders of the Ancient World
• Seven metals of antiquity
• Seven days in the week
• Seven Seas
• Seven Sages
• Seven champions that fought Thebes
• Seven hills of Rome and Seven Kings of Rome
• Seven Sisters, the daughters of Atlas also known as the Pleiades

Religion and mythology

Judaism

See main article: Significance of numbers in Judaism.

The number seven forms a widespread typological pattern within Hebrew scripture, including:

• Seven days (more precisely yom) of Creation, leading to the seventh day or Sabbath (Genesis 1)
• Seven-fold vengeance visited on upon Cain for the killing of Abel (Genesis 4:15)
• Seven pairs of every clean animal loaded onto the ark by Noah (Genesis 7:2)
• Seven years of plenty and seven years of famine in Pharaoh's dream (Genesis 41)
• Seventh son of Jacob, Gad, whose name means good luck (Genesis 46:16)
• Seven times bullock's blood is sprinkled before God (Leviticus 4:6)
• Seven nations God told the Israelites they would displace when they entered the land of Israel (Deuteronomy 7:1)
• Seven days (de jure, but de facto eight days) of the Passover feast (Exodus 13:3–10)
• Seven-branched candelabrum or Menorah (Exodus 25)
• Seven trumpets played by seven priests for seven days to bring down the walls of Jericho (Joshua 6:8)
• Seven things that are detestable to God (Proverbs 6:16–19)
• Seven Pillars of the House of Wisdom (Proverbs 9:1)
• Seven archangels in the deuterocanonical Book of Tobit (12:15)

References to the number seven in Jewish knowledge and practice include:

• Seven divisions of the weekly readings or aliyah of the Torah
• Seven aliyot on Shabbat
• Seven blessings recited under the chuppah during a Jewish wedding ceremony
• Seven days of festive meals for a Jewish bride and groom after their wedding, known as Sheva Berachot or Seven Blessings
• Seven Ushpizzin prayers to the Jewish patriarchs during the holiday of Sukkot

Christianity

Following the tradition of the Hebrew Bible, the New Testament likewise uses the number seven as part of a typological pattern:

• Seven loaves multiplied into seven basketfuls of surplus (Matthew 15:32–37)
• Seven demons were driven out of Mary Magdalene (Luke 8:2)
• Seven last sayings of Jesus on the cross
• Seven men of honest report, full of the Holy Ghost and wisdom (Acts 6:3)
• Seven Spirits of God, Seven Churches and Seven Seals in the Book of Revelation

References to the number seven in Christian knowledge and practice include:

• Seven Gifts of the Holy Spirit
• Seven Corporal Acts of Mercy and Seven Spiritual Acts of Mercy
• Seven deadly sins: lust, gluttony, greed, sloth, wrath, envy, and pride, and seven terraces of Mount Purgatory
• Seven Virtues: chastity, temperance, charity, diligence, kindness, patience, and humility
• Seven Joys and Seven Sorrows of the Virgin Mary
• Seven Sleepers of Christian myth
• Seven Sacraments in the Catholic Church (though some traditions assign a different number)

Islam

References to the number seven in Islamic knowledge and practice include:

• Seven ayat in Surah al-Fatiha, the first chapter of the holy Qur'an
• Seven circumambulations of Muslim pilgrims around the Kaaba in Mecca during the Hajj and the Umrah
• Seven walks between Al-Safa and Al-Marwah performed Muslim pilgrims during the Hajj and the Umrah
• Seven doors to hell (for heaven the number of doors is eight)
• Seven heavens (plural of sky) mentioned in Qur'an
• Night Journey to the Seventh Heaven, (reported ascension to heaven to meet God) Isra' and Mi'raj in Surah Al-Isra'.
• Seventh day naming ceremony held for babies
• Seven enunciators of divine revelation (nāṭiqs) according to the celebrated Fatimid Ismaili dignitary Nasir Khusraw
• Circle Seven Koran, the holy scripture of the Moorish Science Temple of America
• Seven earth as mentioned in the Quran
• Seven children of Muhammad
• Seven years of abundance and seven of drought in Egypt during the time of Yusuf (Joseph) as mentioned in the Quran.

Hinduism

References to the number seven in Hindu knowledge and practice include:

• Seven worlds in the universe and seven seas in the world in Hindu cosmology
• Seven sages or Saptarishi and their seven wives or Sapta Matrka in Hinduism
• Seven Chakras in eastern philosophy
• Seven stars in a constellation called "Saptharishi Mandalam" in Indian astronomy
• Seven promises, or Saptapadi, and seven circumambulations around a fire at Hindu weddings
• Seven virgin goddesses or Saptha Kannimar worshipped in temples in Tamil Nadu, India^{[52] [53]}
• Seven hills at Tirumala known as Yedu Kondalavadu in Telugu, or ezhu malaiyan in Tamil, meaning "Sevenhills God"
• Seven steps taken by the Buddha at birth
• Seven divine ancestresses of humankind in Khasi mythology
• Seven octets or Saptak Swaras in Indian Music as the basis for Ragas compositions
• Seven Social Sins listed by Mahatma Gandhi

Eastern tradition

Other references to the number seven in Eastern traditions include:

• Seven Lucky Gods or gods of good fortune in Japanese mythology
• Seven-Branched Sword in Japanese mythology
• Seven Sages of the Bamboo Grove in China
• Seven minor symbols of yang in Taoist yin-yang

Other references

Other references to the number seven in traditions from around the world include:

• The number seven had mystical and religious significance in Mesopotamian culture by the 22nd century BCE at the latest. This was likely because in the Sumerian sexagesimal number system, dividing by seven was the first division which resulted in infinitely repeating fractions.^[54]
• Seven palms in an Egyptian Sacred Cubit
• Seven ranks in Mithraism
• Seven hills of Istanbul
• Seven islands of Atlantis
• Seven Cherokee clans
• Seven lives of cats in Iran and German and Romance language-speaking cultures^[55]
• Seven fingers on each hand, seven toes on each foot and seven pupils in each eye of the Irish epic hero Cúchulainn
• Seventh sons will be werewolves in Galician folklore, or the son of a woman and a werewolf in other European folklores
• Seventh sons of a seventh son will be magicians with special powers of healing and clairvoyance in some cultures, or vampires in others
• Seven prominent legendary monsters in Guaraní mythology
• Seven gateways traversed by Inanna during her descent into the underworld
• Seven Wise Masters, a cycle of medieval stories
• Seven sister goddesses or fates in Baltic mythology called the Deivės Valdytojos.^[56]
• Seven legendary Cities of Gold, such as Cibola, that the Spanish thought existed in South America
• Seven years spent by Thomas the Rhymer in the faerie kingdom in the eponymous British folk tale
• Seven-year cycle in which the Queen of the Fairies pays a tithe to Hell (or possibly Hel) in the tale of Tam Lin
• Seven Valleys, a text by the Prophet-Founder Bahá'u'lláh in the Bahá'í faith
• Seven superuniverses in the cosmology of Urantia^[57]
• Seven, the sacred number of Yemaya^[58]
• Seven holes representing eyes (سبع عيون) in an Assyrian evil eye bead – though occasionally two, and sometimes nine ^[59]

See also

• Diatonic scale (7 notes)
• Seven colors in the rainbow
• Seven continents
• Seven liberal arts
• Seven Wonders of the Ancient World
• Seven days of the Week
• Septenary (numeral system)
• Year Seven (School)
• Se7en (disambiguation)
• Sevens (disambiguation)
• One-seventh area triangle
• Z with stroke (Ƶ)
• List of highways numbered 7

References

• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group (1987): 70–71

Notes and References

1. [Carl Benjamin Boyer|Carl B. Boyer]
2. 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. 24.67
3. Aamulehti: Opetushallitus harkitsee numero 7 viivan palauttamista . Eeva Törmänen . September 8, 2011 . Tekniikka & Talous . fi . September 9, 2011 . https://web.archive.org/web/20110917083226/http://www.tekniikkatalous.fi/viihde/aamulehti+opetushallitus+harkitsee+numero+7+viivan+palauttamista/a682831 . September 17, 2011 . dead .
4. http://www.adu.by/modules.php?name=News&file=article&sid=858 "Education writing numerals in grade 1."
5. http://www.pour-enfants.fr/jeux-imprimer/apprendre/les-chiffres/ecrire-les-chiffres.png "Example of teaching materials for pre-schoolers"
6. "Nenosen seiska" teki paluun: Tiesitkö, mistä poikkiviiva on peräisin? . Elli Harju . August 6, 2015 . Iltalehti . fi.
7. Web site: Μαθηματικά Α' Δημοτικού . el . Mathematics for the First Grade . Ministry of Education, Research, and Religions . May 7, 2018 . 33.
8. Web site: Weisstein. Eric W.. Double Mersenne Number. 2020-08-06. mathworld.wolfram.com. en.
9. Web site: Sloane's A088165 : NSW primes . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation . 2016-06-01.
10. Web site: Sloane's A050918 : Woodall primes . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation . 2016-06-01.
11. Web site: Sloane's A088054 : Factorial primes . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation . 2016-06-01.
12. Web site: Sloane's A031157 : Numbers that are both lucky and prime . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation . 2016-06-01.
13. Web site: Sloane's A035497 : Happy primes . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation . 2016-06-01.
14. Web site: Sloane's A003173 : Heegner numbers . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation . 2016-06-01.
15. Web site: Weisstein . Eric W. . Heptagon . 2020-08-25 . mathworld.wolfram.com . en.
16. Web site: Weisstein . Eric W. . 7 . 2020-08-07 . mathworld.wolfram.com . en.
17. 2023-01-09.
18. 2016-06-01.
19. Book: Cohen . Henri . Number Theory Volume I: Tools and Diophantine Equations . . 2007 . 978-0-387-49922-2 . 1st . . 239 . 312–314 . Consequences of the Hasse–Minkowski Theorem . 10.1007/978-0-387-49923-9 . 493636622 . 1119.11001.
20. 2024-02-03.
21. Book: Heyden . Anders . Computer Vision – ECCV 2002: 7th European Conference on Computer Vision, Copenhagen, Denmark, May 28–31, 2002. Proceedings. Part II . Sparr . Gunnar . Nielsen . Mads . Johansen . Peter . 2003-08-02 . Springer . 978-3-540-47967-3 . 661 . en . A frieze pattern can be classified into one of the 7 frieze groups....
22. Book: Grünbaum, Branko . Branko Grünbaum . Shephard, G. C. . G.C. Shephard . registration . Tilings and Patterns . Section 1.4 Symmetry Groups of Tilings . W. H. Freeman and Company . New York . 1987 . 40–45 . 10.2307/2323457 . 2323457 . 0-7167-1193-1 . 13092426 . 119730123 .
23. 2023-01-30 .
24. Branko . Grünbaum . Branko Grünbaum . Geoffrey . Shepard . G.C. Shephard . Tilings by Regular Polygons . November 1977 . . 50 . 5 . Taylor & Francis, Ltd.. 231 . 10.2307/2689529 . 2689529 . 123776612 . 0385.51006 .
25. Web site: Jardine . Kevin . Shield - a 3.7.42 tiling . Imperfect Congruence . 2023-01-09 . 3.7.42 as a unit facet in an irregular tiling.
26. Branko . Grünbaum . Branko Grünbaum . Geoffrey . Shepard . G.C. Shephard . Tilings by Regular Polygons . November 1977 . . 50 . 5 . Taylor & Francis, Ltd.. 229–230 . 10.2307/2689529 . 2689529 . 123776612 . 0385.51006 .
27. Book: Dallas, Elmslie William . Elmslie William Dallas . https://books.google.com/books?id=y4BaAAAAcAAJ&pg=PA134 . The Elements of Plane Practical Geometry . Part II. (VII): Of the Circle, with its Inscribed and Circumscribed Figures − Equal Division and the Construction of Polygons . John W. Parker & Son, West Strand . London . 1855 . 134 .

    "...It will thus be found that, including the employment of the same figures, there are seventeen different combinations of regular polygons by which this may be effected; namely, —

    When three polygons are employed, there are ten ways; viz., 6,6,6 – 3.7.42 — 3,8,24 – 3,9,18 — 3,10,15 — 3,12,12 — 4,5,20 — 4,6,12 — 4,8,8 — 5,5,10.

    With four polygons there are four ways, viz., 4,4,4,4 — 3,3,4,12 — 3,3,6,6 — 3,4,4,6.

    With five polygons there are two ways, viz., 3,3,3,4,4 — 3,3,3,3,6.

    With six polygons one way — all equilateral triangles [[[:File:Regular polygons meeting at vertex 6 3 3 3 3 3 3.svg|3.3.3.3.3.3]] ]."
    Note: the only four other configurations from the same combinations of polygons are: 3.4.3.12, (3.6)², 3.4.6.4, and 3.3.4.3.4.

28. Poonen . Bjorn . Bjorn Poonen . Rubinstein . Michael . The Number of Intersection Points Made by the Diagonals of a Regular Polygon . SIAM Journal on Discrete Mathematics . 11 . 1 . . Philadelphia . 1998 . 135–156 . 10.1137/S0895480195281246 . math/9508209 . 1612877 . 0913.51005 . 8673508 .
29. Book: Coxeter, H. S. M. . H. S. M. Coxeter . The Beauty of Geometry: Twelve Essays . https://archive.org/details/beautyofgeometry0000coxe/page/52/mode/2up . registration . Chapter 3: Wythoff's Construction for Uniform Polytopes . Dover Publications . Mineola, NY . 1999 . 326–339 . 9780486409191 . 41565220 . 227201939 . 0941.51001 .
30. Book: Grünbaum, Branko . Branko Grünbaum . Shephard, G. C. . G.C. Shephard . registration . Tilings and Patterns . Section 2.1: Regular and uniform tilings . W. H. Freeman and Company . New York . 1987 . 62–64 . 10.2307/2323457 . 0-7167-1193-1 . 13092426 . 2323457 . 119730123 .
31. Book: Grünbaum, Branko . Branko Grünbaum . Shephard, G. C. . G.C. Shephard . registration . Tilings and Patterns . Section 2.9 Archimedean and uniform colorings . W. H. Freeman and Company . New York . 1987 . 102–107 . 10.2307/2323457 . 0-7167-1193-1 . 13092426 . 2323457 . 119730123 .
32. 2023-01-09 .
33. Branko . Grünbaum . Branko Grünbaum . Geoffrey . Shepard . G.C. Shephard . Tilings by Regular Polygons . November 1977 . . 50 . 5 . Taylor & Francis, Ltd.. 236 . 10.2307/2689529 . 2689529 . 123776612 . 0385.51006 .
34. Book: Tomaž . Pisanski . Brigitte . Servatius . Tomaž Pisanski . Brigitte Servatius . Configurations from a Graphical Viewpoint . Section 1.1: Hexagrammum Mysticum . https://link.springer.com/chapter/10.1007/978-0-8176-8364-1_5 . 1 . . Birkhäuser Advanced Texts . Boston, MA . 2013 . 5–6 . 978-0-8176-8363-4 . 811773514 . 10.1007/978-0-8176-8364-1 . 1277.05001 .
35. Cross products of vectors in higher dimensional Euclidean spaces . William S. . Massey . William S. Massey . The American Mathematical Monthly . 90 . 10 . . December 1983 . 697–701 . 10.2307/2323537 . 2323537 . 43318100 . 0532.55011 . 2023-02-23 . 2021-02-26 . https://web.archive.org/web/20210226011747/https://pdfs.semanticscholar.org/1f6b/ff1e992f60eb87b35c3ceed04272fb5cc298.pdf . dead .
36. Baez . John C. . John Baez . The Octonions . Bulletin of the American Mathematical Society . 39 . 2 . . 152–153 . 2002 . 10.1090/S0273-0979-01-00934-X . 1886087. 586512 . free .
37. Book: Stacey, Blake C. . A First Course in the Sporadic SICs . 2021 . Springer . 978-3-030-76104-2 . Cham, Switzerland . 2–4 . 1253477267.
38. Behrens . M. . Hill . M. . Hopkins . M. J. . Mahowald . M. . 2020 . Detecting exotic spheres in low dimensions using coker J . Journal of the London Mathematical Society . . 101 . 3 . 1173 . 1708.06854 . 10.1112/jlms.12301 . 4111938 . 119170255 . 1460.55017.
39. 2023-02-23.
40. Tumarkin . Pavel . Felikson . Anna . On d-dimensional compact hyperbolic Coxeter polytopes with d + 4 facets . Transactions of the Moscow Mathematical Society . 69 . American Mathematical Society (Translation) . Providence, R.I. . 2008 . 105–151 . 10.1090/S0077-1554-08-00172-6 . free . 2549446 . 37141102 . 1208.52012 .
41. Book: Antoni. F. de. COMPSTAT: Proceedings in Computational Statistics, 7th Symposium held in Rome 1986. Lauro. N.. Rizzi. A.. 2012-12-06. Springer Science & Business Media. 978-3-642-46890-2. 13. en. ...every catastrophe can be composed from the set of so called elementary catastrophes, which are of seven fundamental types..
42. Web site: Weisstein. Eric W.. Dice. 2020-08-25. mathworld.wolfram.com. en.
43. Web site: Millennium Problems Clay Mathematics Institute . 2020-08-25 . www.claymath.org.
44. Web site: 2013-12-15 . Poincaré Conjecture Clay Mathematics Institute . https://web.archive.org/web/20131215120130/http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture . 2013-12-15 . 2020-08-25.
45. Book: Wells, D. . The Penguin Dictionary of Curious and Interesting Numbers . registration . . London . 1987 . 171–174 . 0-14-008029-5 . 39262447 . 118329153 .
46. 2024-04-02 .
47. Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 82
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