Theta

Theta (uppercase Θ or ϴ, lowercase θ (which resembles digit 0 with horizontal line) or ϑ; Ancient Greek: θῆτα, thē̂ta, [tʰɛ̂ːta]; Modern: θήτα, thī́ta, [ˈθita]; UK /ˈθiːtə/, US /ˈθeɪtə/) is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth . In the system of Greek numerals it has a value of 9.

Greek

The alphabet on a black figure vessel, with a point-and-circle theta.

In Ancient Greek, θ represented the aspirated voiceless dental plosive /t̪ʰ/, but in Modern Greek it represents the voiceless dental fricative /θ/.

Forms

In its archaic form, θ was written as a cross within a circle (as in the Etruscan or ), and later, as a line or point within a circle ( or ).

The form ϑ was retained by Unicode as U+03D1 ϑ "GREEK THETA SYMBOL" ("=script theta"), separate from U+03B8 θ "GREEK SMALL LETTER THETA". For the purpose of writing Greek text, the two can be font variants of a single character, but θ and ϑ are also used as distinct symbols in technical and mathematical contexts.

Latin

The name ARAÐÐOVNA on a Gaulish tombstone.

In the Latin script used for the Gaulish language, theta developed into the tau gallicum, conventionally transliterated as Ð, although the bar extends across the centre of the letter. The phonetic value of the tau gallicum is thought to have been [t͡s].

Cyrillic

The early Cyrillic letter fita (Ѳ, ѳ) developed from θ. This letter existed in the Russian alphabet until the 1918 Russian orthography reform.

International Phonetic Alphabet

In the International Phonetic Alphabet, [θ] represents the voiceless dental fricative, as in thick or thin. It does not represent the consonant in the, which is the voiced dental fricative.

Mathematics and science

Lower case

The lower-case letter θ is used as a symbol for:

• A plane angle in geometry.
• A special function of several complex variables.
• One of the Chebyshev functions in prime number theory.
• The potential temperature in meteorology.
• The score of a test taker in item response theory.
• Theta Type Replication: a type of bacterial DNA replication specific to circular chromosomes.
• Threshold value of an artificial neuron.
• A Bayer designation letter applied to a star in a constellation; usually the eighth star so labelled but not necessarily the eighth brightest as viewed from Earth.
• The parameter frequently used in writing the likelihood function (Bayes theorem).
• The Watterson estimator for the population mutation rate in population genetics.
• Indicates a minimum optimum integration level determined by the intersection of GG and LL schedules. The GG-LL schedules are a tool used in analyzing the potential benefits of a country pegging their domestic currency to a foreign currency.
• The reserve ratio of banks in economic models.
• The ordinal collapsing function developed by Solomon Feferman^[1]

Upper case

The upper-case letter Θ is used as a symbol for:

• Quantity or temperature, by SI standard (in italics).
• Dimensionless temperature in transport phenomena.
• An asymptotically tight bound in the analysis of algorithms (big O notation).
• A certain ordinal number in set theory.
• Pentaquarks, exotic baryons in particle physics.
• A brain signal frequency (beta, alpha, theta, delta) ranging from 4–8 Hz.
• One of the variables known as "Greeks" in finance, representing the lost-value rate of an option over time.
• A variable indicating temperature difference in heat transfer.
• In its archaic form () theta is used as a symbol for Earth.

Greek mathematics

The common usage of theta in mathematical problems began with the Greeks. They logically chose this Greek symbol since it was the next in their alphabet not yet used and was easily legible when recreating for multiple uses.

Archaic crossed forms of theta are seen in the wheel letters of Linear A, Linear B, and Egyptian hieroglyphics, the mathematical tensor product, exclusive disjunction, and direct sum operators, the identification symbol for the Earth (as already mentioned), and the Celtic cross. The circumpunct, or circled dot, is seen in a wide range of places.

Symbolism

θ (θάνατος, death) in a mosaic

• In ancient times, Tau was used as a symbol for life or resurrection, whereas the eighth letter of the Greek alphabet, theta, was considered the symbol of death.

According to Porphyry of Tyros, the Egyptians used an X within a circle as a symbol of the soul; having a value of nine, it was used as a symbol for Ennead. Johannes Lydus says that the Egyptians used a symbol for Kosmos in the form of theta, with a fiery circle representing the world, and a snake spanning the middle representing Agathos Daimon (literally: good spirit).^[2]

The Egyptians also used the symbol of a point within a circle (, the sun disc) to represent the sun, which might be a possible origin of its use as the Sun's astrological glyph. It is worthwhile to note that θῆτα (theta) has the same numerical value in isopsephy as Ηλιος (Helios): 318.

Abbreviation

In classical Athens, it was used as an abbreviation for the Greek θάνατος (thanatos, “death”) and as it vaguely resembles a human skull, theta was used as a warning symbol of death, in the same way that skull and crossbones are used in modern times. It survives on potsherds used by Athenians when voting for the death penalty.^[2] Petrus de Dacia in a document from 1291 relates the idea that theta was used to brand criminals as empty ciphers, and the branding rod was affixed to the crossbar spanning the circle.^[3] For this reason, use of the number theta was sometimes avoided where the connotation was felt to be unlucky—the mint marks of some Late Imperial Roman coins famously have the sum ΔΕ or ΕΔ (delta and epsilon, that is 4 and 5) substituted as a euphemism where a Θ (9) would otherwise be expected.

Character encodings

• Greek Theta

Character	Θ		θ		ϑ		ϴ		ᶿ
Unicode name	GREEK CAPITAL LETTER THETA		GREEK SMALL LETTER THETA		GREEK THETA SYMBOL		GREEK CAPITAL THETA SYMBOL		MODIFIER LETTER SMALL THETA
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	920	U+0398	952	U+03B8	977	U+03D1	1012	U+03F4	7615	U+1DBF
UTF-8	206 152	CE 98	206 184	CE B8	207 145	CF 91	207 180	CF B4	225 182 191	E1 B6 BF
Numeric character reference	Θ	Θ	θ	θ	ϑ	ϑ	ϴ	ϴ	ᶿ	ᶿ
Named character reference	Θ		θ		ϑ
DOS Greek	135	87	159	9F
DOS Greek-2	172	AC	226	E2
Windows 1253	200	C8	232	E8
TeX	\Theta		\theta		\vartheta

• Coptic Thethe

Character	Ⲑ		ⲑ
Unicode name	COPTIC CAPITAL LETTER THETHE		COPTIC SMALL LETTER THETHE
Encodings	decimal	hex	decimal	hex
Unicode	11408	U+2C90	11409	U+2C91
UTF-8	226 178 144	E2 B2 90	226 178 145	E2 B2 91
Numeric character reference	Ⲑ	Ⲑ	ⲑ	ⲑ

• Cyrillic Fita

Character	Ѳ		ѳ
Unicode name	CYRILLIC CAPITAL LETTER FITA		CYRILLIC SMALL LETTER FITA
Encodings	decimal	hex	decimal	hex
Unicode	1138	U+0472	1139	U+0473
UTF-8	209 178	D1 B2	209 179	D1 B3
Numeric character reference	Ѳ	Ѳ	ѳ	ѳ

• Mathematical Theta

Character	𝚯		𝛉		𝚹		𝛝
Unicode name	MATHEMATICAL BOLD CAPITAL THETA		MATHEMATICAL BOLD SMALL THETA		MATHEMATICAL BOLD CAPITAL THETA SYMBOL		MATHEMATICAL BOLD THETA SYMBOL
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	120495	U+1D6AF	120521	U+1D6C9	120505	U+1D6B9	120541	U+1D6DD
UTF-8	240 157 154 175	F0 9D 9A AF	240 157 155 137	F0 9D 9B 89	240 157 154 185	F0 9D 9A B9	240 157 155 157	F0 9D 9B 9D
UTF-16	55349 57007	D835 DEAF	55349 57033	D835 DEC9	55349 57017	D835 DEB9	55349 57053	D835 DEDD
Numeric character reference	𝚯	𝚯	𝛉	𝛉	𝚹	𝚹	𝛝	𝛝

Character	𝛩		𝜃		𝛳		𝜗
Unicode name	MATHEMATICAL ITALIC CAPITAL THETA		MATHEMATICAL ITALIC SMALL THETA		MATHEMATICAL ITALIC CAPITAL THETA SYMBOL		MATHEMATICAL ITALIC THETA SYMBOL
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	120553	U+1D6E9	120579	U+1D703	120563	U+1D6F3	120599	U+1D717
UTF-8	240 157 155 169	F0 9D 9B A9	240 157 156 131	F0 9D 9C 83	240 157 155 179	F0 9D 9B B3	240 157 156 151	F0 9D 9C 97
UTF-16	55349 57065	D835 DEE9	55349 57091	D835 DF03	55349 57075	D835 DEF3	55349 57111	D835 DF17
Numeric character reference	𝛩	𝛩	𝜃	𝜃	𝛳	𝛳	𝜗	𝜗

Character	𝜣		𝜽		𝜭		𝝑
Unicode name	MATHEMATICAL BOLD ITALIC CAPITAL THETA		MATHEMATICAL BOLD ITALIC SMALL THETA		MATHEMATICAL BOLD ITALIC CAPITAL THETA SYMBOL		MATHEMATICAL BOLD ITALIC CAPITAL THETA SYMBOL
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	120611	U+1D723	120637	U+1D73D	120621	U+1D72D	120657	U+1D751
UTF-8	240 157 156 163	F0 9D 9C A3	240 157 156 189	F0 9D 9C BD	240 157 156 173	F0 9D 9C AD	240 157 157 145	F0 9D 9D 91
UTF-16	55349 57123	D835 DF23	55349 57149	D835 DF3D	55349 57133	D835 DF2D	55349 57169	D835 DF51
Numeric character reference	𝜣	𝜣	𝜽	𝜽	𝜭	𝜭	𝝑	𝝑

Character	𝝝		𝝷		𝚹		𝞋
Unicode name	MATHEMATICAL SANS-SERIF BOLD CAPITAL THETA		MATHEMATICAL SANS-SERIF BOLD SMALL THETA		MATHEMATICAL SANS-SERIF BOLD CAPITAL THETA SYMBOL		MATHEMATICAL SANS-SERIF BOLD THETA SYMBOL
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	120669	U+1D75D	120695	U+1D777	120505	U+1D6B9	120715	U+1D78B
UTF-8	240 157 157 157	F0 9D 9D 9D	240 157 157 183	F0 9D 9D B7	240 157 154 185	F0 9D 9A B9	240 157 158 139	F0 9D 9E 8B
UTF-16	55349 57181	D835 DF5D	55349 57207	D835 DF77	55349 57017	D835 DEB9	55349 57227	D835 DF8B
Numeric character reference	𝝝	𝝝	𝝷	𝝷	𝚹	𝚹	𝞋	𝞋

Character	𝞗		𝞱		𝞠		𝟅
Unicode name	MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL THETA		MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL THETA		MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL THETA SYMBOL		MATHEMATICAL SANS-SERIF BOLD ITALIC THETA SYMBOL
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	120727	U+1D797	120753	U+1D7B1	120736	U+1D7A0	120773	U+1D7C5
UTF-8	240 157 158 151	F0 9D 9E 97	240 157 158 177	F0 9D 9E B1	240 157 158 160	F0 9D 9E A0	240 157 159 133	F0 9D 9F 85
UTF-16	55349 57239	D835 DF97	55349 57265	D835 DFB1	55349 57248	D835 DFA0	55349 57285	D835 DFC5
Numeric character reference	𝞗	𝞗	𝞱	𝞱	𝞠	𝞠	𝟅	𝟅

These characters are used only as mathematical symbols. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.

See also

• Ѳ, ѳ—Fita, a letter of the early Cyrillic alphabet derived from the Greek theta
• ʘ—Bilabial click
• Voiceless dental fricative

Notes

1. http://math.stanford.edu/~feferman/papers/id-saga.pdf
2. Barry, Kieren (1999), The Greek Qabalah: Alphabetic Mysticism and Numerology in the Ancient World, Samuel Weiser, p. 73, ISBN 1-57863-110-6.
3. Kaplan, Robert (2000), The Nothing that Is: A Natural History of Zero, Oxford; New York: Oxford University Press, p. 66, Bibcode:2000tnti.book.....K, ISBN 0-19-514237-3.