7

From HandWiki
Short description: Integer number 7
This article is about the number. For the years, see 7 BC and AD 7. For other uses, see 7 (disambiguation) and No. 7 (disambiguation).
Short description: Natural number
← 6 7 8 →
-1 0 1 2 3 4 5 6 7 8 9 →
Cardinalseven
Ordinal7th
(seventh)
Numeral systemseptenary
Factorizationprime
Prime4th
Divisors1, 7
Greek numeralΖ´
Roman numeralVII, vii
Greek prefixhepta-/hept-
Latin prefixseptua-
Binary1112
Ternary213
Quaternary134
Quinary125
Senary116
Octal78
Duodecimal712
Hexadecimal716
Vigesimal720
Base 36736
Greek numeralZ, ζ
Amharic፯
Arabic, Kurdish, Persian٧
Sindhi, Urdu۷
Bengali৭
Chinese numeral七, 柒
Devanāgarī७
Telugu౭
Tamil௭
Hebrewז
Khmer៧
Thai๗
Kannada೭
Malayalam൭

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

SevenGlyph.svg

In the beginning, Indians wrote 7 more or less in one stroke as a curve that looks like an uppercase ⟨J⟩ vertically inverted (ᒉ). The western Ghubar Arabs' 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 Arabs 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.

Digital77.svg

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.

Sevens.svg

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 TextFigs078.svg.

Hand Written 7.svg

Most people in Continental Europe,[3] Indonesia,[4] and some in Britain, Ireland, and Canada, as well as Latin America, write 7 with a line in the middle ("7̵"), 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,[5] France,[6] Italy, Belgium, the Netherlands, Finland,[7] Romania, Germany, Greece,[8] and Hungary.[citation needed]

Mathematics

Seven, the fourth prime number, is not only a Mersenne prime (since 23 − 1 = 7) but also a double Mersenne prime since the exponent, 3, is itself a Mersenne prime.[9] It is also a Newman–Shanks–Williams prime,[10] a Woodall prime,[11] a factorial prime,[12] a Harshad number, a lucky prime,[13] a happy number (happy prime),[14] a safe prime (the only Mersenne safe prime), a Leyland prime of the second kind and the fourth Heegner number.[15]

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]
Seven of eight semiregular tilings are Wythoffian, the only exception is the elongated triangular tiling.[30] Seven of nine uniform colorings of the square tiling are also Wythoffian, and between the triangular tiling and square tiling, there are seven non-Wythoffian uniform colorings of a total twenty-one that belong to regular tilings (all hexagonal tiling uniform colorings are 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]
Graph of the probability distribution of the sum of two six-sided dice
Also, 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.[44][45]
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.[46] On the other hand, such figures with rank n + 3 mirrors exist in dimensions 4, 5, 6 and 8; not in 7.[47] Hypercompact polytopes with lowest possible rank of n + 2 mirrors exist up through the 17th dimension, where there is a single solution as well.[48]
  • There are seven fundamental types of catastrophes.[49]
  • When rolling two standard six-sided dice, seven has a 6 in 62 (or 1/6) probability of being rolled (1–6, 6–1, 2–5, 5–2, 3–4, or 4–3), the greatest of any number.[50] The opposite sides of a standard six-sided dice always add to 7.
  • The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000.[51] Currently, six of the problems remain unsolved.[52]

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
Radix 1 5 10 15 20 25 50 75 100 125 150 200 250 500 1000 10000 100000 1000000
x7 1 5 137 217 267 347 1017 1357 2027 2367 3037 4047 5057 13137 26267 411047 5643557 113333117

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.[53] For example, 1/7 = 0.142857 142857... and 2/7 = 0.285714 285714....

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 = 89+5/7; here 5 is the remainder, and would correspond to number 7 in the ranking of the ascending sequence. So in this case, 628 ÷ 7 = 89.714285. Another example, 5238 ÷ 7 = 748+2/7, hence the remainder is 2, and this corresponds to number 2 in the sequence. In this case, 5238 ÷ 7 = 748.285714.

In science

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[54][55]
  • When guessing numbers 1–10, the number 7 is most likely to be picked[56]
  • 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).[57] In Pythagorean numerology the number 7 means spirituality.

References from classical antiquity to the number seven include:

Religion and mythology

Judaism

Main page: Unsolved: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 ayat in surat al-Fatiha, the first book 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 (S. 65:12)
  • Night Journey to the Seventh Heaven, (reported ascension to heaven to meet God) Isra' and Mi'raj of the Qur'an and 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[58]
  • Circle Seven Koran, the holy scripture of the Moorish Science Temple of America
  • Seven earth as mentioned in the Quran[clarification needed]
  • Seven children of Muhammad

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 Hindu mythology
  • 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 [59][60]
  • 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:

  • 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.[61]
  • 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[62]
  • 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.[63]
  • 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[64]
  • Seven, the sacred number of Yemaya[65]
  • Seven holes representing eyes (سبع عيون) in an Assyrian evil eye bead – though occasionally two, and sometimes nine [66]

See also

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" (in fi). Tekniikka & Talous. http://www.tekniikkatalous.fi/viihde/aamulehti+opetushallitus+harkitsee+numero+7+viivan+palauttamista/a682831. Retrieved September 9, 2011. 
  4. ↑ "Mengapa orang Indonesia menambahkan garis kecil pada penulisan angka tujuh (7)?" (in id). Quora. https://id.quora.com/Mengapa-orang-Indonesia-menambahkan-garis-kecil-pada-penulisan-angka-tujuh-7. 
  5. ↑ "Education writing numerals in grade 1." (Russian)
  6. ↑ "Example of teaching materials for pre-schoolers"(French)
  7. ↑ Elli Harju (August 6, 2015). ""Nenosen seiska" teki paluun: Tiesitkö, mistä poikkiviiva on peräisin?" (in fi). Iltalehti. http://www.iltalehti.fi/uutiset/2015080620139397_uu.shtml. 
  8. ↑ "Μαθηματικά Α' Δημοτικού" (in el). Ministry of Education, Research, and Religions. p. 33. http://ebooks.edu.gr/modules/document/file.php/DSDIM-A102/%CE%94%CE%B9%CE%B4%CE%B1%CE%BA%CF%84%CE%B9%CE%BA%CF%8C%20%CE%A0%CE%B1%CE%BA%CE%AD%CF%84%CE%BF/%CE%92%CE%B9%CE%B2%CE%BB%CE%AF%CE%BF%20%CE%9C%CE%B1%CE%B8%CE%B7%CF%84%CE%AE/10-0007-02_Mathimatika_A-Dim_BM-1.pdf. 
  9. ↑ Weisstein, Eric W.. "Double Mersenne Number" (in en). https://mathworld.wolfram.com/DoubleMersenneNumber.html. 
  10. ↑ "Sloane's A088165 : NSW primes". OEIS Foundation. https://oeis.org/A088165. 
  11. ↑ "Sloane's A050918 : Woodall primes". OEIS Foundation. https://oeis.org/A050918. 
  12. ↑ "Sloane's A088054 : Factorial primes". OEIS Foundation. https://oeis.org/A088054. 
  13. ↑ "Sloane's A031157 : Numbers that are both lucky and prime". OEIS Foundation. https://oeis.org/A031157. 
  14. ↑ "Sloane's A035497 : Happy primes". OEIS Foundation. https://oeis.org/A035497. 
  15. ↑ "Sloane's A003173 : Heegner numbers". OEIS Foundation. https://oeis.org/A003173. 
  16. ↑ Heyden, Anders; Sparr, Gunnar; Nielsen, Mads; Johansen, Peter (2003-08-02) (in en). Computer Vision - ECCV 2002: 7th European Conference on Computer Vision, Copenhagen, Denmark, May 28-31, 2002. Proceedings. Part II. Springer. pp. 661. ISBN 978-3-540-47967-3. https://books.google.com/books?id=4yCqCAAAQBAJ&q=seven+frieze+groups&pg=PA661. "A frieze pattern can be classified into one of the 7 frieze groups..." 
  17. ↑ 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. OCLC 13092426. https://archive.org/details/isbn_0716711931. 
  18. ↑ Sloane, N. J. A., ed. "Sequence A004029 (Number of n-dimensional space groups.)". OEIS Foundation. https://oeis.org/A004029. Retrieved 2023-01-30. 
  19. ↑ Sloane, N. J. A., ed. "Sequence A000040 (The prime numbers)". OEIS Foundation. https://oeis.org/A000040. Retrieved 2023-02-01. 
  20. ↑ Weisstein, Eric W.. "Heptagon" (in en). https://mathworld.wolfram.com/Heptagon.html. 
  21. ↑ Weisstein, Eric W.. "7" (in en). https://mathworld.wolfram.com/7.html. 
  22. ↑ Sloane, N. J. A., ed. "Sequence A000566 (Heptagonal numbers (or 7-gonal numbers))". OEIS Foundation. https://oeis.org/A000566. Retrieved 2023-01-09. 
  23. ↑ Sloane, N. J. A., ed. "Sequence A003215". OEIS Foundation. https://oeis.org/A003215. Retrieved 2016-06-01. 
  24. ↑ Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons". Mathematics Magazine (Taylor & Francis, Ltd.) 50 (5): 231. doi:10.2307/2689529. http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf. 
  25. ↑ Jardine, Kevin. "Shield - a 3.7.42 tiling". http://gruze.org/tilings/3_7_42_shield.  3.7.42 as a unit facet in an irregular tiling.
  26. ↑ Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons". Mathematics Magazine (Taylor & Francis, Ltd.) 50 (5): 229–230. doi:10.2307/2689529. http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf. 
  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. https://books.google.com/books?id=y4BaAAAAcAAJ&pg=PA134. 
    "...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". SIAM Journal on Discrete Mathematics (Philadelphia: Society for Industrial and Applied Mathematics) 11 (1): 135–156. doi:10.1137/S0895480195281246. https://math.mit.edu/~poonen/papers/ngon.pdf. 
  29. ↑ Coxeter, H. S. M. (1999). "Chapter 3: Wythoff's Construction for Uniform Polytopes". The Beauty of Geometry: Twelve Essays. Mineola, NY: Dover Publications. pp. 326–339. ISBN 9780486409191. OCLC 41565220. https://archive.org/details/beautyofgeometry0000coxe/page/52/mode/2up. 
  30. ↑ Grünbaum, Branko; Shephard, G. C. (1987). "Section 2.1: Regular and uniform tilings". Tilings and Patterns. New York: W. H. Freeman and Company. pp. 62–64. doi:10.2307/2323457. ISBN 0-7167-1193-1. OCLC 13092426. https://archive.org/details/isbn_0716711931. 
  31. ↑ Grünbaum, Branko; Shephard, G. C. (1987). "Section 2.9 Archimedean and uniform colorings". Tilings and Patterns. New York: W. H. Freeman and Company. pp. 102–107. doi:10.2307/2323457. ISBN 0-7167-1193-1. OCLC 13092426. https://archive.org/details/isbn_0716711931. 
  32. ↑ Sloane, N. J. A., ed. "Sequence A068600 (Number of n-uniform tilings having n different arrangements of polygons about their vertices.)". OEIS Foundation. https://oeis.org/A068600. Retrieved 2023-01-09. 
  33. ↑ Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons". Mathematics Magazine (Taylor & Francis, Ltd.) 50 (5): 236. doi:10.2307/2689529. http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf. 
  34. ↑ 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. https://link.springer.com/chapter/10.1007/978-0-8176-8364-1_5. 
  35. ↑ Pisanski, Tomaž; Servatius, Brigitte (2013). "Chapter 5.3: Classical Configurations". Configurations from a Graphical Viewpoint. Birkhäuser Advanced Texts (1 ed.). Boston, MA: Birkhäuser. pp. 170–173. doi:10.1007/978-0-8176-8364-1. ISBN 978-0-8176-8363-4. OCLC 811773514. https://link.springer.com/chapter/10.1007/978-0-8176-8364-1_5. 
  36. ↑ Szilassi, Lajos (1986). "Regular toroids". Structural Topology 13: 74. http://www-iri.upc.es/people/ros/StructuralTopology/ST13/st13-06-a3-ocr.pdf. 
  37. ↑ Császár, Ákos (1949). "A polyhedron without diagonals". Acta Scientiarum Mathematicarum (Szeged) 13: 140–142. http://www.diale.org/pdf/csaszar.pdf. 
  38. ↑ Sloane, N. J. A., ed. "Sequence A004031 (Number of n-dimensional crystal systems.)". OEIS Foundation. https://oeis.org/A004031. Retrieved 2023-01-30. 
  39. ↑ Wang, Gwo-Ching; Lu, Toh-Ming (2014). "Crystal Lattices and Reciprocal Lattices". RHEED Transmission Mode and Pole Figures (1 ed.). New York: Springer Publishing. pp. 8–9. doi:10.1007/978-1-4614-9287-0_2. ISBN 978-1-4614-9286-3. https://link.springer.com/chapter/10.1007/978-1-4614-9287-0_2. 
  40. ↑ Sloane, N. J. A., ed. "Sequence A256413 (Number of n-dimensional Bravais lattices)". OEIS Foundation. https://oeis.org/A256413. Retrieved 2023-01-30. 
  41. ↑ Messer, Peter W. (2002). "Closed-Form Expressions for Uniform Polyhedra and Their Duals". Discrete & Computational Geometry (Springer) 27 (3): 353–355; 372–373. doi:10.1007/s00454-001-0078-2. https://link.springer.com/content/pdf/10.1007/s00454-001-0078-2.pdf. 
  42. ↑ Massey, William S. (December 1983). "Cross products of vectors in higher dimensional Euclidean spaces". The American Mathematical Monthly (Taylor & Francis, Ltd) 90 (10): 697–701. doi:10.2307/2323537. https://pdfs.semanticscholar.org/1f6b/ff1e992f60eb87b35c3ceed04272fb5cc298.pdf. Retrieved 2023-02-23. 
  43. ↑ Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society (American Mathematical Society) 39 (2): 152–153. doi:10.1090/S0273-0979-01-00934-X. http://math.ucr.edu/home/baez/octonions/. 
  44. ↑ 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 (London Mathematical Society) 101 (3): 1173. doi:10.1112/jlms.12301. https://onlinelibrary.wiley.com/doi/abs/10.1112/jlms.12301. 
  45. ↑ Sloane, N. J. A., ed. "Sequence A001676 (Number of h-cobordism classes of smooth homotopy n-spheres.)". OEIS Foundation. https://oeis.org/A001676. Retrieved 2023-02-23. 
  46. ↑ Tumarkin, Pavel; Felikson, Anna (2008). "On d-dimensional compact hyperbolic Coxeter polytopes with d + 4 facets". Transactions of the Moscow Mathematical Society (Providence, R.I.: American Mathematical Society (Translation)) 69: 105–151. doi:10.1090/S0077-1554-08-00172-6. https://www.ams.org/journals/mosc/2008-69-00/S0077-1554-08-00172-6/S0077-1554-08-00172-6.pdf. 
  47. ↑ Tumarkin, Pavel (2007). "Compact hyperbolic Coxeter n-polytopes with n + 3 facets". The Electronic Journal of Combinatorics 14 (1): 1–36 (R69). doi:10.37236/987. https://www.combinatorics.org/ojs/index.php/eljc/article/view/v14i1r69/pdf. 
  48. ↑ Tumarkin, P. V. (2004). "Hyperbolic Coxeter N-Polytopes with n+2 Facets". Mathematical Notes 75 (6): 848–854. doi:10.1023/b:matn.0000030993.74338.dd. 
  49. ↑ Antoni, F. de; Lauro, N.; Rizzi, A. (2012-12-06) (in en). COMPSTAT: Proceedings in Computational Statistics, 7th Symposium held in Rome 1986. Springer Science & Business Media. pp. 13. ISBN 978-3-642-46890-2. https://books.google.com/books?id=3L_sCAAAQBAJ&q=seven+fundamental+types+of+catastrophes&pg=PA13. "...every catastrophe can be composed from the set of so called elementary catastrophes, which are of seven fundamental types." 
  50. ↑ Weisstein, Eric W.. "Dice" (in en). https://mathworld.wolfram.com/Dice.html. 
  51. ↑ "Millennium Problems | Clay Mathematics Institute". http://www.claymath.org/millennium-problems. 
  52. ↑ "Poincaré Conjecture | Clay Mathematics Institute". 2013-12-15. http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture. 
  53. ↑ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 82
  54. ↑ Gonzalez, Robbie (4 December 2014). "Why Do People Love The Number Seven?" (in en-us). https://gizmodo.com/why-do-people-love-the-number-seven-so-much-1666353786. 
  55. ↑ Bellos, Alex. "The World's Most Popular Numbers [Excerpt"]. https://www.scientificamerican.com/article/most-popular-numbers-grapes-of-math/. 
  56. ↑ 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. https://www.researchgate.net/publication/232582800. Retrieved 20 February 2022. 
  57. ↑ "Number symbolism - 7". https://www.britannica.com/topic/number-symbolism/7. 
  58. ↑ "Nāṣir-i Khusraw", An Anthology of Philosophy in Persia (I.B.Tauris), 2001, doi:10.5040/9780755610068.ch-008, ISBN 978-1-84511-542-5, http://dx.doi.org/10.5040/9780755610068.ch-008, retrieved 2020-11-17 
  59. ↑ Rajarajan, R.K.K. (2020). "Peerless Manifestations of Devī.". Carcow Indological Studies (Cracow, Poland) XXII.1: 221–243. doi:10.12797/CIS.22.2020.01.09. https://www.academia.edu/44394046. 
  60. ↑ Rajarajan, R.K.K. (2020). "Sempiternal "Pattiṉi": Archaic Goddess of the vēṅkai-tree to Avant-garde Acaṉāmpikai". Studia Orientalia Electronica (Helsinki, Finland) 8 (1): 120–144. doi:10.23993/store.84803. https://www.academia.edu/43913587. 
  61. ↑ The Origin of the Mystical Number Seven in Mesopotamian Culture: Division by Seven in the Sexagesimal Number System
  62. ↑ "Encyclopædia Britannica "Number Symbolism"". Britannica.com. http://www.britannica.com/EBchecked/topic/1086220/number-symbolism/248164/7. 
  63. ↑ Klimka, Libertas (2012-03-01). "Senosios baltų mitologijos ir religijos likimas". Lituanistica 58 (1). doi:10.6001/lituanistica.v58i1.2293. ISSN 0235-716X. 
  64. ↑ "Chapter I. The Creative Thesis of Perfection by William S. Sadler, Jr. - Urantia Book - Urantia Foundation". 17 August 2011. http://www.urantia.org/study-master-universe/chapter-i-creative-thesis-perfection. 
  65. ↑ Yemaya. Santeria Church of the Orishas. Retrieved 25 November 2022
  66. ↑ Ergil, Leyla Yvonne (2021-06-10). "Turkey's talisman superstitions: Evil eyes, pomegranates and more". https://www.dailysabah.com/turkey/expat-corner/turkeys-talisman-superstitions-evil-eyes-pomegranates-and-more. 

References

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



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