7

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.[1] 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,[2] Indonesia,[3] and some in Britain, Ireland, and Canada, as well as Latin America, write 7 with a line in 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 the two can appear similar when written in certain styles of handwriting. This form is used in official handwriting rules for primary school in Russia, Ukraine, Bulgaria, Poland, other Slavic countries,[4] France,[5] Italy, Belgium, the Netherlands, Finland,[6] Romania, Germany, Greece,[7] and Hungary.

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.[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 prime of the second kind and the fourth Heegner number.[14]

• Seven is the lowest natural number that cannot be represented as the sum of the squares of three integers. (See Lagrange's four-square theorem#Historical development.)

• Seven is the aliquot sum of one number, the cubic number 8 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.

• 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.[15] These are related to the 17 wallpaper groups whose transformations and isometries repeat two-dimensional patterns in the plane.[16][17] The seventh indexed prime number is seventeen.[18]

• A seven-sided shape is a heptagon.[19] 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.[20] Figurate numbers representing heptagons are called heptagonal numbers.[21] 7 is also a centered hexagonal number.[22]

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).[23][24] This is also one of twenty-one such configurations from seventeen combinations of polygons, that features the largest and smallest polygons possible.[25][26]

• In Wythoff's kaleidoscopic constructions, seven distinct generator points that lie on mirror edges of a three-sided Schwarz triangle are used to create most uniform tilings and polyhedra; an eighth point lying on all three mirrors is technically degenerate, reserved to represent snub forms only.[27]

Seven of eight semiregular tilings are Wythoffian, the only exception is the elongated triangular tiling.[28] 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).[29]

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.[30][31]

• The Fano plane is the smallest possible finite projective plane with 7 points and 7 lines such that every line contains 3 points and 3 lines cross every point.[32] With group order 168 = 2³·3·7, this plane holds 35 total triples of points where 7 are collinear and another 28 are non-collinear, whose incidence graph is the 3-regular bipartate Heawood graph with 14 vertices and 21 edges.[33] This graph embeds in three dimensions as the Szilassi polyhedron, the simplest toroidal polyhedron alongside its dual with 7 vertices, the Császár polyhedron.[34][35]

• In tridimensional space there are seven crystal systems and fourteen Bravais lattices which classify under seven lattice systems, six of which are shared with the seven crystal systems.[36][37][38] There are also collectively seventy-seven Wythoff symbols that represent all uniform figures in three dimensions.[39]

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

• The seventh dimension is the only dimension aside from the familiar three where a vector cross product can be defined.[40] This is related to the octonions over the imaginary subspace Im(O) in 7-space whose commutator between two octonions defines this vector product, wherein the Fano plane describes the multiplicative algebraic structure of the unit octonions {e₀, e₁, e₂, ..., e₇}, with e₀ an identity element.[41]

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.[42][43]

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.[44] On the other hand, such figures with rank n + 3 mirrors exist in dimensions 4, 5, 6 and 8; not in 7.[45] Hypercompact polytopes with lowest possible rank of n + 2 mirrors exist up through the 17th dimension, where there is a single solution as well.[46]

• There are seven fundamental types of catastrophes.[47]

• 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.[48] 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.[49] Currently, six of the problems remain unsolved.[50]

• Seven colors in a rainbow: ROYGBIV
• Seven Continents
• Seven Seas
• Seven climes
• The neutral pH balance
• Number of music notes in a scale
• Number of spots most commonly found on ladybugs
• Atomic number for nitrogen
• Number of diatomic elements
• Seven basic crystal systems

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).[55] In Pythagorean numerology the number 7 means spirituality.

References from classical antiquity to the number seven include:

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

Wikimedia Commons has media related to 7 (number).

Look up seven in Wiktionary, the free dictionary.

• 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

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