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

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,^{[ 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 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.^{[ 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. ^[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 = 2³ making it the base of the 7-aliquot tree; it is also, therefore, the sum-of-divisors of only 4, its prime index. Furthermore, the smallest number with seven divisors is 64 = 8². ^[15]
• The seventh triangular number is the second perfect number 28 = 7 × 4, ^[16] which precedes 6. ^[17] In decimal representation, the reciprocal of 7 repeats six digits (as 0.142857), ^{[18] [19]} whose sum when cycling back to 1 is equal to 28. On the other hand, 7 is the number of partitions of 5, ^[20] a value n which yields the third perfect number 496 for 2^{n − 1}(2ⁿ − 1), by the Euclid-Euler theorem.
• 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. ^[21] These are related to the 17 wallpaper groups whose transformations and isometries repeat two-dimensional patterns in the plane. ^{[22] [23]} The seventh indexed prime number is seventeen. ^[24]
• As a consequence of Fermat's little theorem and Euler's criterion, all cubes are congruent to 0, 1, or 6 modulo 7.
• A seven-sided shape is a heptagon. ^[25] 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. ^[26] Figurate numbers representing heptagons are called heptagonal numbers. ^[27] 7 is also a centered hexagonal number. ^[28]

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). ^{[29] [30]} This is also one of twenty-one such configurations from seventeen combinations of polygons, that features the largest and smallest polygons possible. ^{[31] [32]}

Otherwise, for any regular n-sided polygon, the maximum number of intersecting diagonals (other than through its center) is at most 7. ^[33]

• 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. ^[34]

Seven of eight semiregular tilings are Wythoffian (the only exception is the elongated triangular tiling), where there exist three tilings that are regular, all of which are Wythoffian. ^[35] 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). ^[36]

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. ^{[37] [38]}

• 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. ^[39] 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. ^[40] This graph embeds in three dimensions as the Szilassi polyhedron, the simplest toroidal polyhedron alongside its dual with 7 vertices, the Császár polyhedron. ^{[41] [42]}
• In three-dimensional 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. ^{[43] [44] [45]} There are also collectively seventy-seven Wythoff symbols that represent all uniform figures in three dimensions. ^[46]

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. ^[47] 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. ^[48]

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. ^{[49] [50]}

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. ^[51] On the other hand, such figures with rank n + 3 mirrors exist in dimensions 4, 5, 6 and 8; not in 7. ^[52] Hypercompact polytopes with lowest possible rank of n + 2 mirrors exist up through the 17th dimension, where there is a single solution as well. ^[53]

• There are seven fundamental types of catastrophes. ^[54]
• The positive definite quadratic integer 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. ^{[55] [56]}
• 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. ^[57] 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. ^[58] Currently, six of the problems remain unsolved. ^[59]

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. ^[60] 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

• Seven colors in a rainbow
• 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

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 ^{[61] [62]}
• When guessing numbers 1–10, the number 7 is most likely to be picked ^[63]
• 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). ^[64] 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

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 ayat in surat al-Fatiha, the first chapter 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 ^[65]
• 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
• Seven years of abundance and seven of drought in Egypt during the time of Yusuf (Joseph) as mentioned in the Koran. ^[66]

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 ^{[67] [68]}
• 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. ^[69]
• 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 ^[70]
• 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. ^[71]
• 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 ^[72]
• Seven, the sacred number of Yemaya ^[73]
• Seven holes representing eyes (سبع عيون) in an Assyrian evil eye bead – though occasionally two, and sometimes nine ^[74]

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's A003173 : Heegner numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
15. Sloane, N. J. A. (ed.). "SequenceA000005(d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-05.
16. Sloane, N. J. A. (ed.). "SequenceA000217(Triangular numbers: a(n) as the binomial(n+1,2) equal to n*(n+1)/2 or 0 + 1 + 2 + ... + n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-02.
17. Sloane, N. J. A. (ed.). "SequenceA000396(Perfect numbers k: k is equal to the sum of the proper divisors of k.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-02.
18. 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.
19. Sloane, N. J. A. (ed.). "SequenceA060283(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.
20. Sloane, N. J. A. (ed.). "SequenceA000041(a(n) is the number of partitions of n (the partition numbers).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-04-02.
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. Sloane, N. J. A. (ed.). "SequenceA000040(The prime numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-02-01.
25. Weisstein, Eric W. "Heptagon". mathworld.wolfram.com. Retrieved 2020-08-25.
26. Weisstein, Eric W. "7". mathworld.wolfram.com. Retrieved 2020-08-07.
27. Sloane, N. J. A. (ed.). "SequenceA000566(Heptagonal numbers (or 7-gonal numbers))". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-01-09.
28. Sloane, N. J. A. (ed.). "SequenceA003215". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-06-01.
29. 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.
30. 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.
31. 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.
32. 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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References

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