3

3 is the natural number following 2|2 and preceding 4|4.

In mathematics

 * Three is approximately π (actually closer to 3.14159) when doing rapid engineering guesses or estimates. The same is true if one wants a rough-and-ready estimate of e, which is actually approximately 2.71828.
 * Three is the first odd prime number, and the second smallest prime. It is both the first Fermat prime (22 n + 1) and the first Mersenne prime (2n &minus; 1), the only number that is both, as well as the first lucky prime. However, it is the second Sophie Germain prime, the second Mersenne prime exponent, the second factorial prime (2! + 1), the second Lucas prime, the second Stern prime.
 * Three is the first unique prime due to the properties of its reciprocal.
 * Three is the aliquot sum of 4.
 * Three is the third Heegner number.
 * According to Pythagoras and the Pythagorean school, the number 3, which they called triad, is the noblest of all digits, as it is the only number to equal the sum of all the terms below it, and the only number whose sum with those below equals the product of them and itself.
 * Three is the second triangular number and it is the only prime triangular number. Three is the only prime which is one less than a perfect square. Any other number which is n2 &minus; 1 for some integer n is not prime, since it is (n &minus; 1)(n + 1). This is true for 3 as well, but in its case one of the factors is 1.
 * Three non-collinear points determine a plane and a circle.
 * Three is the fourth Fibonacci number. In the Perrin sequence, however, 3 is both the zeroth and third Perrin numbers.
 * Three is the fourth open meandric number.
 * Vulgar fractions with 3 in the denominator have a single digit repeating sequences in their decimal expansions, (.000..., .333..., .666...)
 * A natural number is divisible by three if the sum of its digits in base 10 is divisible by 3. For example, the number 21 is divisible by three (3 times 7) and the sum of its digits is 2 + 1 = 3. Because of this, the reverse of any number that is divisible by three (or indeed, any permutation of its digits) is also divisible by three. For instance, 1368 and its reverse 8631 are both divisible by three (and so are 1386, 3168, 3186, 3618, etc.). This works in base 10 and in any positional numeral system whose base divided by three leaves a remainder of one (bases 4, 7, 10, etc.).
 * A triangle is the only figure which, if all endpoints have hinges, will never change its shape unless the sides themselves are bent.
 * 3 is the smallest prime of a Mersenne prime power tower 3, 7, 127, 170141183460469231731687303715884105727. It is not known whether any more of the terms are prime.
 * Three of the five regular polyhedra have triangular faces — the tetrahedron, the octahedron, and the icosahedron. Also, three of the five regular polyhedra have vertices where three faces meet — the tetrahedron, the hexahedron (cube), and the dodecahedron. Furthermore, only three different types of polygons comprise the faces of the five regular polyhedra — the triangle, the quadrilateral, and the pentagon.
 * There are only three distinct 4×4 panmagic squares.
 * Only three tetrahedral numbers are also perfect squares.
 * The first number, according to the pythagoreans, and the first male number.
 * The first number, according to Proclus, being the first number such that n2 is greater than 2n.
 * The trisection of the angle was one of the three famous problems of antiquity.
 * 3 is the second triangular number.
 * Gauss proved that every integer is the sum of at most 3 triangular numbers.
 * Gauss proved that for any prime number p (with the sole exception of 3) the product of its primitive roots is ≡ 1 (mod p).
 * Any number not in the form of 4n(8m+7) is the sum of 3 squares.

In numeral systems
It is frequently noted by historians of numbers that early counting systems often relied on the three-patterned concept of "One- Two- Many" to describe counting limits. In other words, in their own language equivalent way, early peoples had a word to describe the quantities of one and two, but any quantity beyond this point was simply denoted as "Many". As an extension to this insight, it can also be noted that early counting systems appear to have had limits at the numerals 2, 3, and 4. References to counting limits beyond these three indices do not appear to prevail as consistently in the historical record.

Evolution of the glyph
Three is the largest number still written with as many lines as the number represents. (The Ancient Romans usually wrote 4 as IIII, but this was almost entirely replaced by the subtractive notation IV in the Middle Ages.) To this day 3 is written as three lines in Roman and Chinese numerals. This was the way the Brahmin Indians wrote it, and the Gupta made the three lines more curved. The Nagari started rotating the lines clockwise and ending each line with a slight downward stroke on the right. Eventually they made these strokes connect with the lines below, and evolved it to a character that looks very much like a modern 3 with an extra stroke at the bottom. It was the Western Ghubar Arabs who finally eliminated the extra stroke and created our modern 3. (The "extra" stroke, however, was very important to the Eastern Arabs, and they made it much larger, while rotating the strokes above to lie along a horizontal axis, and to this day Eastern Arabs write a 3 that looks like a mirrored 7 with ridges on its top line): ٣

While the shape of the 3 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender. In some French text-figure typefaces, though, it has an ascender instead of a descender.

A common variant of the digit 3 has a flat top. Since this form is sometimes used to prevent people from fraudulently changing a 3 into an 8, it is sometimes called a banker's 3.

In science

 * The Roman numeral III stands for giant star in the Yerkes spectral classification scheme.
 * Three is the atomic number of lithium.
 * We perceive our universe to have three spatial dimensions, but some theories suggest there are more that we're not able to detect, such as string theory.

In religion
Many world religions contain triple deities or concepts of trinity, including:
 * the Christian Holy Trinity
 * the Hindu Trimurti
 * the Hindu Tridevi
 * the Three Jewels of Buddhism
 * the Three Pure Ones of Taoism
 * the Triple Goddess of Wicca

As a lucky or unlucky number
Three (三, or formally: 叁) is considered a good number in Chinese culture because it sounds like the word "alive" (生), compared to four (四), which sounds like the word "death" (死).

Counting to three is common in situations where a group of people wish to perform an action in synchrony: Now, on the count of three, everybody pull! Assuming the counter is proceeding at a uniform rate, the first two counts are necessary to establish the rate, but then everyone can predict when three" will come based on "one" and "two"; this is likely why three is used instead of some other number.

In Vietnam, there is a superstition that considers it bad luck to take a photo with three people in it; it is professed that the person in the middle will die soon.

There is another superstition that it is unlucky to take a third light, that is, to be the third person to light a cigarette from the same match or lighter. This superstition is sometimes asserted to have originated among soldiers in the trenches of the First World War when a sniper might see the first light, take aim on the second and fire on the third.

The phrase "Third time's the charm" refers to the superstition that after two failures in any endeavor, a third attempt is more likely to succeed. This is also sometimes seen in reverse, as in "third man [to do something, presumably forbidden] gets caught".

Luck, especially bad luck, is often said to "come in threes".

In philosophy

 * The three Doshas (weaknesses) and their antidotes are the basis of Ayurvedic medicine in India.
 * Philosophers such as Aquinas, Kant, Hegel, and Charles Sanders have made threefold divisions, or trichotomies, which have been important in their work.
 * Hegel's dialectic of Thesis + Antithesis = Synthesis creates three-ness from two-ness.