7

This article is about the number. For the year, see AD 7. For other uses, see 7 (disambiguation) and No. 7 (disambiguation).

Not to be confused with ⁊.

← 6 7 8 →
−1 0 1 2 3 4 5 6 7 8 9 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	seven
Ordinal	7th (seventh)
Numeral system	septenary
Factorization	prime
Prime	4th
Divisors	1, 7
Greek numeral	Ζ´
Roman numeral	VII, vii
Greek prefix	hepta-/hept-
Latin prefix	septua-
Binary	1112
Ternary	213
Senary	116
Octal	78
Duodecimal	712
Hexadecimal	716
Greek numeral	Z, ζ
Amharic	፯
Arabic, Kurdish, Persian	٧
Sindhi, Urdu	۷
Bengali	৭
Chinese numeral	七, 柒
Devanāgarī	७
Telugu	౭
Tamil	௭
Hebrew	ז
Khmer	៧
Thai	๗
Kannada	೭
Malayalam	൭
Armenian	Է
Babylonian numeral	𒐛
Egyptian hieroglyph	𓐀
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.^{[ citation needed ]}

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 devices use three line segments, but devices made by some Japanese companies such as Sharp and Casio, as well as in the Koreas and Taiwan, 7 is written with four line segments because in those countries, 7 is written with a "hook" on the left, as ① in the following illustration. Further segments can give further variation. For example, Schindler elevators in the United States and Canada installed or modernized from the late 1990s onwards usually use a sixteen segment display and show the digit 7 in a manner more similar to that of handwriting.

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,^{[ citation needed ]} and some in Britain, Ireland, and Canada, as well as Latin America, write 7 with a line through the middle (7), sometimes with the top line crooked. The line through the middle is useful to clearly differentiate the digit from the digit one, as they 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.^{[ citation needed ]}

In mathematics

Seven, the fourth prime number, is not only a Mersenne prime (since ${\displaystyle 2^{3}-1=7}$) 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 Mersenne safe prime), a Leyland number of the second kind ^[14] and Leyland prime of the second kind ^[15] (${\displaystyle 2^{5}-5^{2}}$), and the fourth Heegner number. ^[16] 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. ^[17] 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. ^[18]

7 is the only number D for which the equation 2ⁿ − D = x² has more than two solutions for n and x natural. In particular, the equation 2ⁿ − 7 = x² 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: {1, 3, 5, 7, 11, 15, 33}. ^{[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 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. ^{[29] [30]}

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. ^[31] 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. ^{[32] [33] [34]}

Graph of the probability distribution of the sum of two six-sided dice

The lowest known dimension for an exotic sphere is the seventh dimension. ^{[35] [36]}

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. On the other hand, such figures with rank n + 3 mirrors exist in dimensions 4, 5, 6 and 8; not in 7. ^[37]

There are seven fundamental types of catastrophes. ^[38]

When rolling two standard six-sided dice, seven has a 1 in 6 probability of being rolled, the greatest of any number. ^[39] 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. ^[40] Currently, six of the problems remain unsolved. ^[41]

Basic calculations

Multiplication	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	50	100	1000
7 × x	7	14	21	28	35	42	49	56	63	70	77	84	91	98	105	112	119	126	133	140	147	154	161	168	175	350	700	7000

Division	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
7 ÷ x	7	3.5	2.3	1.75	1.4	1.16	1	0.875	0.7	0.7	0.63	0.583	0.538461	0.5	0.46
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	2.142857

Exponentiation	1	2	3	4	5	6	7	8	9	10	11	12	13
7x	7	49	343	2401	16807	117649	823543	5764801	40353607	282475249	1977326743	13841287201	96889010407
x7	1	128	2187	16384	78125	279936	823543	2097152	4782969	10000000	19487171	35831808	62748517

In decimal

999,999 divided by 7 is exactly 142,857. 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. ^[42] In decimal representation, the reciprocal of 7 repeats six digits (as 0.142857), ^{[43] [44]} whose sum when cycling back to 1 is equal to 28.

In science

• 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
• Seven basic colours of the rainbow

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 ^{[45] [46]}
• When guessing numbers 1–10, the number 7 is most likely to be picked ^[47]
• 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). ^[48] 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 who 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

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 lampstands in The Vision of John on Patmos by Julius Schnorr von Carolsfeld, 1860

• 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 is the first number to be mentioned in Quran ^[49]
• Seven ayat in Surah al-Fatiha, the first chapter of the Quran
• 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
• Seven heavens (plural of sky) mentioned in Quran ^[50]
• 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 Takbir at the first rak'a of the Eid prayers
• Seven pebble is used during the ritual of throwing of the jamarāt
• Seven sins that doom one to Hell ^[51]
• Seven Ajwa dates in the morning to avoid being poisoned or bewitched that day ^[52]
• seven fat cows eaten up by seven skinny ones; and seven green ears of grain and seven others dry In the dream of the king of egypt ^[53]
• Seven types of mutlaq water ^[54]
• Seven point of body during sujud ^[55]
• Seven line of sentence in adhan
• Seven stages of life ^[56]
• The first hajj since the Hijrah at the 7th year of hijri calendar ^[57]
• The seventh month of Hijri calendar (Rajab) is one of the four scared month in Islam
• Seven enunciators of divine revelation (nāṭiqs) according to the celebrated Fatimid Ismaili dignitary Nasir Khusraw ^[58]
• Seven layer of earth as mentioned in the Quran ^[59]
• Seven children of Muhammad ^[60]
• Seven years of abundance and seven of drought in Egypt during the time of Yusuf (Joseph) as mentioned in the Quran. ^[61]

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 ^{[62] [63]}
• 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:

The Seven Lucky Gods in Japanese mythology

• 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:

• Seven colors in a rainbow
• 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. ^[64]
• 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 ^[65]
• 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. ^[66]
• 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 ^[67]
• Seven, the sacred number of Yemaya ^[68]
• Seven holes representing eyes (سبع عيون) in an Assyrian evil eye bead – though occasionally two, and sometimes nine ^[69]

See also

• Mathematics portal

• 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

Notes

1. Carl B. Boyer, A History of Mathematics (1968) p.52, 2nd edn.
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. Eeva Törmänen (September 8, 2011). "Aamulehti: Opetushallitus harkitsee numero 7 viivan palauttamista". Tekniikka & Talous (in Finnish). Archived from the original on September 17, 2011. Retrieved September 9, 2011.
4. "Education writing numerals in grade 1." Archived 2008-10-02 at the Wayback Machine (Russian)
5. "Example of teaching materials for pre-schoolers"(French)
6. Elli Harju (August 6, 2015). ""Nenosen seiska" teki paluun: Tiesitkö, mistä poikkiviiva on peräisin?". Iltalehti (in Finnish).
7. "Μαθηματικά Α' Δημοτικού" [Mathematics for the First Grade](PDF) (in Greek). Ministry of Education, Research, and Religions. p. 33. Retrieved May 7, 2018.
8. Weisstein, Eric W. "Double Mersenne Number". mathworld.wolfram.com. Retrieved 2020-08-06.
9. "Sloane's A088165 : NSW primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
10. "Sloane's A050918 : Woodall primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
11. "Sloane's A088054 : Factorial primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
12. "Sloane's A031157 : Numbers that are both lucky and prime". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
13. "Sloane's A035497 : Happy primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
14. Sloane, N. J. A. (ed.). "SequenceA045575(Leyland numbers of the second kind)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
15. Sloane, N. J. A. (ed.). "SequenceA123206(Leyland prime numbers of the second kind)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
16. "Sloane's A003173 : Heegner numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
17. Weisstein, Eric W. "Heptagon". mathworld.wolfram.com. Retrieved 2020-08-25.
18. Weisstein, Eric W. "7". mathworld.wolfram.com. Retrieved 2020-08-07.
19. Cohen, Henri (2007). "Consequences of the Hasse–Minkowski Theorem". Number Theory Volume I: Tools and Diophantine Equations. Graduate Texts in Mathematics. Vol. 239 (1st ed.). Springer. pp. 312–314. doi:10.1007/978-0-387-49923-9. ISBN   978-0-387-49922-2. OCLC   493636622. Zbl   1119.11001.
20. Sloane, N. J. A. (ed.). "SequenceA116582(Numbers from Bhargava's 33 theorem.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-03.
21. Heyden, Anders; Sparr, Gunnar; Nielsen, Mads; Johansen, Peter (2003-08-02). Computer Vision – ECCV 2002: 7th European Conference on Computer Vision, Copenhagen, Denmark, May 28–31, 2002. Proceedings. Part II. Springer. p. 661. ISBN   978-3-540-47967-3. A frieze pattern can be classified into one of the 7 frieze groups...
22. Grünbaum, Branko; Shephard, G. C. (1987). "Section 1.4 Symmetry Groups of Tilings". Tilings and Patterns . New York: W. H. Freeman and Company. pp. 40–45. doi:10.2307/2323457. ISBN   0-7167-1193-1. JSTOR   2323457. OCLC   13092426. S2CID   119730123.
23. Sloane, N. J. A. (ed.). "SequenceA004029(Number of n-dimensional space groups.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-01-30.
24. Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine . 50 (5). Taylor & Francis, Ltd.: 231. doi:10.2307/2689529. JSTOR   2689529. S2CID   123776612. Zbl   0385.51006.
25. Jardine, Kevin. "Shield - a 3.7.42 tiling". Imperfect Congruence. Retrieved 2023-01-09. 3.7.42 as a unit facet in an irregular tiling.
26. Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine . 50 (5). Taylor & Francis, Ltd.: 229–230. doi:10.2307/2689529. JSTOR   2689529. S2CID   123776612. Zbl   0385.51006.
27. Dallas, Elmslie William (1855). "Part II. (VII): Of the Circle, with its Inscribed and Circumscribed Figures − Equal Division and the Construction of Polygons". The Elements of Plane Practical Geometry. London: John W. Parker & Son, West Strand. p. 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 [ 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; Rubinstein, Michael (1998). "The Number of Intersection Points Made by the Diagonals of a Regular Polygon" (PDF). SIAM Journal on Discrete Mathematics. 11 (1). Philadelphia: Society for Industrial and Applied Mathematics: 135–156. arXiv: math/9508209 . doi:10.1137/S0895480195281246. MR   1612877. S2CID   8673508. Zbl   0913.51005.
29. Sloane, N. J. A. (ed.). "SequenceA068600(Number of n-uniform tilings having n different arrangements of polygons about their vertices.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-01-09.
30. Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine . 50 (5). Taylor & Francis, Ltd.: 236. doi:10.2307/2689529. JSTOR   2689529. S2CID   123776612. Zbl   0385.51006.
31. Pisanski, Tomaž; Servatius, Brigitte (2013). "Section 1.1: Hexagrammum Mysticum". Configurations from a Graphical Viewpoint. Birkhäuser Advanced Texts (1 ed.). Boston, MA: Birkhäuser. pp. 5–6. doi:10.1007/978-0-8176-8364-1. ISBN   978-0-8176-8363-4. OCLC   811773514. Zbl   1277.05001.
32. Massey, William S. (December 1983). "Cross products of vectors in higher dimensional Euclidean spaces" (PDF). The American Mathematical Monthly. 90 (10). Taylor & Francis, Ltd: 697–701. doi:10.2307/2323537. JSTOR   2323537. S2CID   43318100. Zbl   0532.55011. Archived from the original (PDF) on 2021-02-26. Retrieved 2023-02-23.
33. Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society. 39 (2). American Mathematical Society: 152–153. doi: 10.1090/S0273-0979-01-00934-X . MR   1886087. S2CID   586512.
34. Stacey, Blake C. (2021). A First Course in the Sporadic SICs. Cham, Switzerland: Springer. pp. 2–4. ISBN   978-3-030-76104-2. OCLC   1253477267.
35. 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). London Mathematical Society: 1173. arXiv: 1708.06854 . doi:10.1112/jlms.12301. MR   4111938. S2CID   119170255. Zbl   1460.55017.
36. Sloane, N. J. A. (ed.). "SequenceA001676(Number of h-cobordism classes of smooth homotopy n-spheres.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-02-23.
37. Tumarkin, Pavel; Felikson, Anna (2008). "On d-dimensional compact hyperbolic Coxeter polytopes with d + 4 facets" (PDF). Transactions of the Moscow Mathematical Society. 69. Providence, R.I.: American Mathematical Society (Translation): 105–151. doi: 10.1090/S0077-1554-08-00172-6 . MR   2549446. S2CID   37141102. Zbl   1208.52012.
38. Antoni, F. de; Lauro, N.; Rizzi, A. (2012-12-06). COMPSTAT: Proceedings in Computational Statistics, 7th Symposium held in Rome 1986. Springer Science & Business Media. p. 13. ISBN   978-3-642-46890-2. ...every catastrophe can be composed from the set of so called elementary catastrophes, which are of seven fundamental types.
39. Weisstein, Eric W. "Dice". mathworld.wolfram.com. Retrieved 2020-08-25.
40. "Millennium Problems | Clay Mathematics Institute". www.claymath.org. Retrieved 2020-08-25.
41. "Poincaré Conjecture | Clay Mathematics Institute". 2013-12-15. Archived from the original on 2013-12-15. Retrieved 2020-08-25.
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References

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