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.
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,[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]
Seven is the aliquot sum of one number, the cubic number8 and is the base of the 7-aliquot tree.
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.
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-uniformKrotenheerdt 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]
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]
The positive definite quadraticinteger matrix representative of all odd numbers contains the set of seven integers: {1, 3, 5, 7, 11, 15, 33} where seven is the middle indexed member.[50][51]
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.[52] The opposite sides of a standard six-sided dice always add to 7.
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.[55] 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 Western culture, seven is consistently listed as people's favorite number[56][57]
When guessing numbers 1–10, the number 7 is most likely to be picked[58]
Seven-year itch, a term that suggests that happiness in a marriage declines after around seven years
Classical antiquityedit
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).[59] 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
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.[63]
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^"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's A003173 : Heegner numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
^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.
^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. Philadelphia: Society for Industrial and Applied Mathematics. 11 (1): 135–156. arXiv:math/9508209. doi:10.1137/S0895480195281246. MR 1612877. S2CID 8673508. Zbl 0913.51005.
^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. S2CID 227201939. Zbl 0941.51001.
^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. JSTOR 2323457. OCLC 13092426. S2CID 119730123.
^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. JSTOR 2323457. OCLC 13092426. S2CID 119730123.
^Császár, Ákos (1949). "A polyhedron without diagonals" (PDF). Acta Scientiarum Mathematicarum (Szeged). 13: 140–142. Archived from the original (PDF) on 2017-09-18.
^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. S2CID 124399480.
^Messer, Peter W. (2002). "Closed-Form Expressions for Uniform Polyhedra and Their Duals" (PDF). Discrete & Computational Geometry. Springer. 27 (3): 353–355, 372–373. doi:10.1007/s00454-001-0078-2. MR 1921559. S2CID 206996937. Zbl 1003.52006.
^Massey, William S. (December 1983). "Cross products of vectors in higher dimensional Euclidean spaces" (PDF). The American Mathematical Monthly. Taylor & Francis, Ltd. 90 (10): 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.
^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. arXiv:1708.06854. doi:10.1112/jlms.12301. MR 4111938. S2CID 119170255. Zbl 1460.55017.
^Tumarkin, Pavel; Felikson, Anna (2008). "On d-dimensional compact hyperbolic Coxeter polytopes with d + 4 facets" (PDF). Transactions of the Moscow Mathematical Society. Providence, R.I.: American Mathematical Society (Translation). 69: 105–151. doi:10.1090/S0077-1554-08-00172-6. MR 2549446. S2CID 37141102. Zbl 1208.52012.
^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. MR 2350459. S2CID 221033082. Zbl 1168.51311.
^Tumarkin, P. V. (2004). "Hyperbolic Coxeter N-Polytopes with n+2 Facets". Mathematical Notes. 75 (6): 848–854. arXiv:math/0301133. doi:10.1023/b:matn.0000030993.74338.dd. MR 2086616. S2CID 15156852. Zbl 1062.52012.
^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.
^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.
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^The Origin of the Mystical Number Seven in Mesopotamian Culture: Division by Seven in the Sexagesimal Number System
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