7: Difference between revisions

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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	_ _...

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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 $2^{3}-1=7$ ) 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] ( $2^{5}-5^{2}$ ), 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]

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	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
7^x	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:

"Number Seven"
by William Sidney Gibson
Read by Ruth Golding for LibriVox

Audio 00:15:59 (full text)

Problems playing this file? See media help.

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

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 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:

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]

Notes

^ Carl B. Boyer, A History of Mathematics (1968) p.52, 2nd edn.
^ 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
^ 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.
^ "Education writing numerals in grade 1." Archived 2008-10-02 at the Wayback Machine(Russian)
^ "Example of teaching materials for pre-schoolers"(French)
^ Elli Harju (August 6, 2015). ""Nenosen seiska" teki paluun: Tiesitkö, mistä poikkiviiva on peräisin?". Iltalehti (in Finnish).
^ "Μαθηματικά Α' Δημοτικού" [Mathematics for the First Grade] (PDF) (in Greek). Ministry of Education, Research, and Religions. p. 33. Retrieved May 7, 2018.
^ Weisstein, Eric W. "Double Mersenne Number". mathworld.wolfram.com. Retrieved 2020-08-06.
^ "Sloane's A088165 : NSW primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
^ "Sloane's A050918 : Woodall primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
^ "Sloane's A088054 : Factorial primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
^ "Sloane's A031157 : Numbers that are both lucky and prime". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
^ "Sloane's A035497 : Happy primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
^ Sloane, N. J. A. (ed.). "Sequence A045575 (Leyland numbers of the second kind)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Sloane, N. J. A. (ed.). "Sequence A123206 (Leyland prime numbers of the second kind)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ "Sloane's A003173 : Heegner numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
^ Weisstein, Eric W. "Heptagon". mathworld.wolfram.com. Retrieved 2020-08-25.
^ Weisstein, Eric W. "7". mathworld.wolfram.com. Retrieved 2020-08-07.
^ 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.
^ Sloane, N. J. A. (ed.). "Sequence A116582 (Numbers from Bhargava's 33 theorem.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-03.
^ 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...
^ 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.
^ Sloane, N. J. A. (ed.). "Sequence A004029 (Number of n-dimensional space groups.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-30.
^ 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.
^ 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.
^ 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.
^ 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.
^ 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.
^ Sloane, N. J. A. (ed.). "Sequence A068600 (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.
^ 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.
^ 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.
^ 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.
^ 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.
^ 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.
^ 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.
^ Sloane, N. J. A. (ed.). "Sequence A001676 (Number of h-cobordism classes of smooth homotopy n-spheres.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-23.
^ 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.
^ 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.
^ Weisstein, Eric W. "Dice". mathworld.wolfram.com. Retrieved 2020-08-25.
^ "Millennium Problems | Clay Mathematics Institute". www.claymath.org. Retrieved 2020-08-25.
^ "Poincaré Conjecture | Clay Mathematics Institute". 2013-12-15. Archived from the original on 2013-12-15. Retrieved 2020-08-25.
^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 82
^ Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin Books. pp. 171–174. ISBN 0-14-008029-5. OCLC 39262447. S2CID 118329153.
^ Sloane, N. J. A. (ed.). "Sequence A060283 (Periodic part of decimal expansion of reciprocal of n-th prime (leading 0's moved to end).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-02.
^ Gonzalez, Robbie (4 December 2014). "Why Do People Love The Number Seven?". Gizmodo. Retrieved 20 February 2022.
^ Bellos, Alex. "The World's Most Popular Numbers [Excerpt]". Scientific American. Retrieved 20 February 2022.
^ Kubovy, Michael; Psotka, Joseph (May 1976). "The predominance of seven and the apparent spontaneity of numerical choices". Journal of Experimental Psychology: Human Perception and Performance. 2 (2): 291–294. doi:10.1037/0096-1523.2.2.291. Retrieved 20 February 2022.
^ "Number symbolism – 7".
^ 2:29
^ 65:12
^ "Sunan an-Nasa'i 3671". sunnah.com. n.d. Retrieved 2024-11-11.
^ "IRSYAD HUKUM SIRI KE-347 : FADILAT MEMAKAN TUJUH BIJI KURMA, ADAKAH KHUSUS BUAT KURMA AJWAH MADINAH SAHAJA ATAU UMUM BUAT SEMUA JENIS KURMA?". muftiwp.gov.my (in Malay). 2019-12-18. Retrieved 2024-11-11.
^ 12:43
^ "Air Mutlak: Maksud, Jenis & Kegunaan". ecentral (in Malay). 2024-08-22. Retrieved 2024-11-11.
^ "AL-KAFI #1261: IS THE NOSE INCLUDED AS THE BODY PARTS OF SUJUD (PROSTRATION IN PRAYER)?". muftiwp.gov.my. 2019-05-17. Retrieved 2024-11-11.
^ "Seven Stages of Life in Islam". islamicity. 2020-11-27. Retrieved 2024-11-11.
^ Wan Abdullah, Wan Ali Akbar (2018). Kurikulum Standard Sekolah Menengah: 3 فنديدين إسلام تیغکتن • pendidikan Islam tingkatan 3 [Secondary School Standard Curriculum: form 3 Islamic education • form 3 islamic education] (in Malay). kementerian pendidikan Malaysia (Malaysian Ministry of Education). ISBN 9789834920814.
^ "Nāṣir-i Khusraw", An Anthology of Philosophy in Persia, I.B.Tauris, pp. 305–361, 2001, doi:10.5040/9780755610068.ch-008, ISBN 978-1-84511-542-5, retrieved 2020-11-17
^ 65:12
^ Ahmad jibriel Mohd Yusof (2015-03-24). "Anak nabi kita ada 7 orang". Retrieved 2024-11-11.
^ Surah Yusuf 12:46
^ Rajarajan, R.K.K. (2020). "Peerless Manifestations of Devī". Carcow Indological Studies (Cracow, Poland). 22 (1): 221–243. doi:10.12797/CIS.22.2020.01.09. S2CID 226326183.
^ Rajarajan, R.K.K. (2020). "Sempiternal "Pattiṉi": Archaic Goddess of the vēṅkai-tree to Avant-garde Acaṉāmpikai". Studia Orientalia Electronica. 8 (1): 120–144. doi:10.23993/store.84803. S2CID 226373749.
^ The Origin of the Mystical Number Seven in Mesopotamian Culture: Division by Seven in the Sexagesimal Number System
^ "Encyclopædia Britannica "Number Symbolism"". Britannica.com. Retrieved 2012-09-07.
^ Klimka, Libertas (2012-03-01). "Senosios baltų mitologijos ir religijos likimas". Lituanistica. 58 (1). doi:10.6001/lituanistica.v58i1.2293. ISSN 0235-716X.
^ "Chapter I. The Creative Thesis of Perfection by William S. Sadler, Jr. – Urantia Book – Urantia Foundation". urantia.org. 17 August 2011.
^ Yemaya. Santeria Church of the Orishas. Retrieved 25 November 2022
^ Ergil, Leyla Yvonne (2021-06-10). "Turkey's talisman superstitions: Evil eyes, pomegranates and more". Daily Sabah. Retrieved 2023-04-05.

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[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

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[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.

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[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.

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[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.

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[54] "Air Mutlak: Maksud, Jenis & Kegunaan". ecentral (in Malay). 2024-08-22. Retrieved 2024-11-11.

[55] "AL-KAFI #1261: IS THE NOSE INCLUDED AS THE BODY PARTS OF SUJUD (PROSTRATION IN PRAYER)?". muftiwp.gov.my. 2019-05-17. Retrieved 2024-11-11.

[56] "Seven Stages of Life in Islam". islamicity. 2020-11-27. Retrieved 2024-11-11.

[57] Wan Abdullah, Wan Ali Akbar (2018). Kurikulum Standard Sekolah Menengah: 3 فنديدين إسلام تیغکتن • pendidikan Islam tingkatan 3 [Secondary School Standard Curriculum: form 3 Islamic education • form 3 islamic education] (in Malay). kementerian pendidikan Malaysia (Malaysian Ministry of Education). ISBN 9789834920814.

[58] "Nāṣir-i Khusraw", An Anthology of Philosophy in Persia, I.B.Tauris, pp. 305–361, 2001, doi:10.5040/9780755610068.ch-008, ISBN 978-1-84511-542-5, retrieved 2020-11-17

[59] 65:12

[60] Ahmad jibriel Mohd Yusof (2015-03-24). "Anak nabi kita ada 7 orang". Retrieved 2024-11-11.

[61] Surah Yusuf 12:46

[62] Rajarajan, R.K.K. (2020). "Peerless Manifestations of Devī". Carcow Indological Studies (Cracow, Poland). 22 (1): 221–243. doi:10.12797/CIS.22.2020.01.09. S2CID 226326183.

[63] Rajarajan, R.K.K. (2020). "Sempiternal "Pattiṉi": Archaic Goddess of the vēṅkai-tree to Avant-garde Acaṉāmpikai". Studia Orientalia Electronica. 8 (1): 120–144. doi:10.23993/store.84803. S2CID 226373749.

[64] The Origin of the Mystical Number Seven in Mesopotamian Culture: Division by Seven in the Sexagesimal Number System

[65] "Encyclopædia Britannica "Number Symbolism"". Britannica.com. Retrieved 2012-09-07.

[66] Klimka, Libertas (2012-03-01). "Senosios baltų mitologijos ir religijos likimas". Lituanistica. 58 (1). doi:10.6001/lituanistica.v58i1.2293. ISSN 0235-716X.

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[68] Yemaya. Santeria Church of the Orishas. Retrieved 25 November 2022

[Ergil_2021-69] Ergil, Leyla Yvonne (2021-06-10). "Turkey's talisman superstitions: Evil eyes, pomegranates and more". Daily Sabah. Retrieved 2023-04-05.

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