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21-552: For example, in the letter y , the descender is the "tail", or that portion of the diagonal line which lies below the v created by the two lines converging. In the letter p , it is the stem reaching down past the ɒ . In most fonts, descenders are reserved for lowercase characters such as g , j , q , p , y , and sometimes f . Some fonts, however, also use descenders for some numerals (typically 3 , 4 , 5 , 7 , and 9 ). Such numerals are called old-style numerals . (Some italic fonts, such as Computer Modern italic , put

42-444: A convex polygon is a polygon that is the boundary of a convex set . This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon (not self-intersecting ). Equivalently, a polygon is convex if every line that does not contain any edge intersects the polygon in at most two points. A strictly convex polygon

63-582: A polygon , a diagonal is a line segment joining any two non-consecutive vertices. Therefore, a quadrilateral has two diagonals, joining opposite pairs of vertices. For any convex polygon , all the diagonals are inside the polygon, but for re-entrant polygons , some diagonals are outside of the polygon. Any n -sided polygon ( n ≥ 3), convex or concave , has n ( n − 3 ) 2 {\displaystyle {\tfrac {n(n-3)}{2}}} total diagonals, as each vertex has diagonals to all other vertices except itself and

84-423: A descender on the numeral 4 but not on any other numerals. Such fonts are not considered old-style.) Some fonts also use descenders for the tails on a few uppercase letters such as J and Q . The parts of characters that extend above the x-height of a font are called ascenders . Descenders are often reduced in small-print typefaces for uses such as newspapers, directories or pocket Bibles to fit more text on

105-411: A page. More radically, on 20 May 1802 Philip Rusher of Banbury patented a new Patent Type with eliminated descenders and shortened ascenders. The type did not prove successful, nor did another use in 1852. The Art Nouveau display typeface Hobo and headline typeface Permanent Headline which also eliminate descenders have both been somewhat popular since. Some early computer displays (for example,

126-572: A single point in the interior, the number of regions that the diagonals divide the interior into is given by For n -gons with n =3, 4, ... the number of regions is This is OEIS sequence A006522. If no three diagonals of a convex polygon are concurrent at a point in the interior, the number of interior intersections of diagonals is given by ( n 4 ) {\displaystyle \textstyle {\binom {n}{4}}} . This holds, for example, for any regular polygon with an odd number of sides. The formula follows from

147-434: Is a convex polygon such that no line contains two of its edges. In a convex polygon, all interior angles are less than or equal to 180 degrees, while in a strictly convex polygon all interior angles are strictly less than 180 degrees. The following properties of a simple polygon are all equivalent to convexity: Additional properties of convex polygons include: Every polygon inscribed in a circle (such that all vertices of

168-411: Is called diagonal. The word diagonal derives from the ancient Greek διαγώνιος diagonios , "from corner to corner" (from διά- dia- , "through", "across" and γωνία gonia , "corner", related to gony "knee"); it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid , and later adopted into Latin as diagonus ("slanting line"). As applied to

189-472: Is to look at the diagonal on the two- torus S xS and observe that it can move off itself by the small motion (θ, θ) to (θ, θ + ε). In general, the intersection number of the graph of a function with the diagonal may be computed using homology via the Lefschetz fixed-point theorem ; the self-intersection of the diagonal is the special case of the identity function. Convex polygon In geometry ,

210-575: The Compukit UK101 ) and printers (for example, the Commodore 4022) restricted the vertical spacing of characters so that there was no space for correct display of descenders. Instead, characters with descenders were displaced vertically upwards so that the bottom of the descender was aligned with the baseline . Contemporary systems that did not have this restriction were described as supporting true descenders . Some type designers have observed

231-565: The fixed points of a mapping F from X to itself may be obtained by intersecting the graph of F with the diagonal. In geometric studies, the idea of intersecting the diagonal with itself is common, not directly, but by perturbing it within an equivalence class . This is related at a deep level with the Euler characteristic and the zeros of vector fields . For example, the circle S has Betti numbers 1, 1, 0, 0, 0, and therefore Euler characteristic 0. A geometric way of expressing this

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252-517: The xth shortest diagonal. As an example, a 5-cube would have the diagonals: Its total number of diagonals is 416. In general, an n-cube has a total of 2 n − 1 ( 2 n − n − 1 ) {\displaystyle 2^{n-1}(2^{n}-n-1)} diagonals. This follows from the more general form of v ( v − 1 ) 2 − e {\displaystyle {\frac {v(v-1)}{2}}-e} which describes

273-401: The fact that each intersection is uniquely determined by the four endpoints of the two intersecting diagonals: the number of intersections is thus the number of combinations of the n vertices four at a time. Although the number of distinct diagonals in a polygon increases as its number of sides increases, the length of any diagonal can be calculated. In a regular n-gon with side length a ,

294-462: The interior of the polyhedron (except for the endpoints on the vertices). The lengths of an n-dimensional hypercube 's diagonals can be calculated by mathematical induction . The longest diagonal of an n-cube is n {\displaystyle {\sqrt {n}}} . Additionally, there are 2 n − 1 ( n x + 1 ) {\displaystyle 2^{n-1}{\binom {n}{x+1}}} of

315-444: The length of the xth shortest distinct diagonal is: This formula shows that as the number of sides approaches infinity, the xth shortest diagonal approaches the length (x+1)a . Additionally, the formula for the shortest diagonal simplifies in the case of x = 1: If the number of sides is even, the longest diagonal will be equivalent to the diameter of the polygon's circumcircle because the long diagonals all intersect each other at

336-585: The polygon's center. Special cases include: A square has two diagonals of equal length, which intersect at the center of the square. The ratio of a diagonal to a side is 2 ≈ 1.414. {\displaystyle {\sqrt {2}}\approx 1.414.} A regular pentagon has five diagonals all of the same length. The ratio of a diagonal to a side is the golden ratio , 1 + 5 2 ≈ 1.618. {\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.618.} A regular hexagon has nine diagonals:

357-399: The seven longer ones equal each other. The reciprocal of the side equals the sum of the reciprocals of a short and a long diagonal. A polyhedron (a solid object in three-dimensional space , bounded by two-dimensional faces ) may have two different types of diagonals: face diagonals on the various faces, connecting non-adjacent vertices on the same face; and space diagonals, entirely in

378-410: The six shorter ones are equal to each other in length; the three longer ones are equal to each other in length and intersect each other at the center of the hexagon. The ratio of a long diagonal to a side is 2, and the ratio of a short diagonal to a side is 3 {\displaystyle {\sqrt {3}}} . A regular heptagon has 14 diagonals. The seven shorter ones equal each other, and

399-535: The total number of face and space diagonals in convex polytopes. Here, v represents the number of vertices and e represents the number of edges. By analogy, the subset of the Cartesian product X × X of any set X with itself, consisting of all pairs (x,x), is called the diagonal, and is the graph of the equality relation on X or equivalently the graph of the identity function from X to X . This plays an important part in geometry; for example,

420-451: The two adjacent vertices, or n  − 3 diagonals, and each diagonal is shared by two vertices. In general, a regular n -sided polygon has ⌊ n − 2 2 ⌋ {\displaystyle \lfloor {\frac {n-2}{2}}\rfloor } distinct diagonals in length, which follows the pattern 1,1,2,2,3,3... starting from a square. In a convex polygon , if no three diagonals are concurrent at

441-399: The “tails” of some Thai glyphs to be analogues to the descenders. (e.g. ฤ ) Other authors have also mentioned pedestals/feet to be analogues to the descenders. (e.g. ฎ , ฏ , ญ and ฐ ) Diagonal In geometry , a diagonal is a line segment joining two vertices of a polygon or polyhedron , when those vertices are not on the same edge . Informally, any sloping line

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