hcju/proofwikips huggingface dataset:
Definition:Rubik's Cube/Facet
Each of the $9$ small squares into which the faces of Rubik's cube is subdivided is called a '''facet'''.
Definition:Symmetric Difference/Definition 5
:$S * T := \set {x: x \in S \oplus x \in T}$
Definition:Rubik's Cube/Corner Facet
Of each of the $9$ facets of each face of Rubik's cube, the $4$ in the corners of the face are called the '''corner facets'''. The set of '''corner facets''' can be denoted $S_C$.
Definition:Rubik's Cube/Edge Facet
Of each of the $9$ facets of each face of Rubik's cube, the $4$ on the edges of the face are called the '''edge facets'''. The set of '''edge facets''' can be denoted $S_E$.
Definition:Rubik's Cube/Center Facet
Of each of the $9$ facets of each face of Rubik's cube, the ones in the centers of the faces are called the '''center facets'''. The set of '''center facets''' can be denoted $S_Z$.
Definition:Trivial Partition/Singleton
The '''singleton partition''' on $S$ is defined as: :$\mathcal P = \left\{{S}\right\}$ That is, it is a partition with only one component.
Definition:Trivial Partition/Partition of Singletons
The '''partition of singletons''' on $S$ is defined as: :$\mathcal P = \set {\set x: x \in S}$ That is, it is a partition such that every component is a singleton.
Definition:Prime Number/Definition 5
A '''prime number''' $p$ is an integer greater than $1$ that has no (positive) divisors less than itself other than $1$.
Definition:Prime-Counting Function/Examples
The values of the prime-counting ($\pi$) function for the first few integers are as follows: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="right" style = "padding: 2px 10px" | $\map \pi n$  |- | align="right" style = "padding: 2px 10px" | $1$ | align="right" style = "padding: 2px 10px" | $0$ |- | align="right" style = "padding: 2px 10px" | $2$ | align="right" style = "padding: 2px 10px" | $1$ |- | align="right" style = "padding: 2px 10px" | $3$ | align="right" style = "padding: 2px 10px" | $2$ |- | align="right" style = "padding: 2px 10px" | $4$ | align="right" style = "padding: 2px 10px" | $2$ |- | align="right" style = "padding: 2px 10px" | $5$ | align="right" style = "padding: 2px 10px" | $3$ |- | align="right" style = "padding: 2px 10px" | $6$ | align="right" style = "padding: 2px 10px" | $3$ |- | align="right" style = "padding: 2px 10px" | $7$ | align="right" style = "padding: 2px 10px" | $4$ |- | align="right" style = "padding: 2px 10px" | $8$ | align="right" style = "padding: 2px 10px" | $4$ |}
Definition:Prime-Counting Function/Examples/16
The value of the prime-counting ($\pi$) function for $16$ is determined as follows. The prime numbers less than $16$ are: :$2, 3, 5, 7, 11, 13$ Hence: :$\map \pi {16} = 6$
Definition:Mapping/Definition 1
A '''mapping''' from $S$ to $T$ is a binary relation on $S \times T$ which associates each element of $S$ with exactly one element of $T$.
Definition:Mapping/Definition 2
A '''mapping $f$ from $S$ to $T$''', denoted $f: S \to T$, is a relation $f = \struct {S, T, G}$, where $G \subseteq S \times T$, such that: :$\forall x \in S: \forall y_1, y_2 \in T: \tuple {x, y_1} \in G \land \tuple {x, y_2} \in G \implies y_1 = y_2$ and :$\forall x \in S: \exists y \in T: \tuple {x, y} \in G$
Definition:Mapping/Definition 3
A '''mapping $f$ from $S$ to $T$''', denoted $f: S \to T$, is a relation $f = \struct {S, T, R}$, where $R \subseteq S \times T$, such that: :$\forall \tuple {x_1, y_1}, \tuple {x_2, y_2} \in f: y_1 \ne y_2 \implies x_1 \ne x_2$ and :$\forall x \in S: \exists y \in T: \tuple {x, y} \in f$
Definition:Mapping/Definition 4
A '''mapping from $S$ to $T$''' is a relation on $S \times T$ which is: :$(1): \quad$ Many-to-one :$(2): \quad$ Left-total, that is, defined for all elements in $S$.
Definition:Countable Set/Also known as
When the terms '''denumerable''' and '''enumerable''' are encountered, they generally mean the same as countably infinite. Some modern pedagogues (for example {{AuthorRef|Vi Hart}} and {{AuthorRef|James Grime}}) use the term '''listable''', but this has yet to catch on.
Definition:Countable Set/Countably Infinite/Cardinality
The cardinality of a countably infinite set is denoted by the symbol $\aleph_0$ ('''aleph null''').
Definition:Countable Set/Countably Infinite/Definition 1
$S$ is '''countably infinite'''  there exists a bijection: :$f: S \to \N$ where $\N$ is the set of natural numbers.
Definition:Countable Set/Countably Infinite/Definition 2
$S$ is '''countably infinite'''  there exists a bijection: :$f: S \to \Z$ where $\Z$ is the set of integers.
Definition:Countable Set/Also defined as
Some sources define a countable set to be what is defined on  as a countably infinite set. That is, they use '''countable''' to describe a set which has ''exactly the same'' cardinality as $\N$. Thus under this criterion $X$ is said to be countable  there exists a bijection from $X$ to $\N$, that is,  $X$ is equivalent to $\N$. However, as the very concept of the term '''countable''' implies that a set '''can be counted''', which, plainly, a finite set can be, it is suggested that this interpretation may be counter-intuitive. Hence, on , the term countable set will be taken in the sense as to include the concept of finite set, and countably infinite will mean a countable set which is specifically ''not'' finite.
Definition:Countable Set/Definition 3
$S$ is '''countable'''  there exists a bijection between $S$ and a subset of $\N$.
Definition:Polish Notation/Formal Definition
Let $\mathcal A$ be an alphabet. Let each $s \in \mathcal A$ be assigned a natural number called its arity. The formal grammar for Polish notation is given by the single bottom-up rule: ::If $s$ has arity $n$ and $\phi_1, \ldots, \phi_n$ are well-formed formulas, then: :::$s \phi_1 \cdots \phi_n$ ::is also a well-formed formula. Notably, in the case where $s$ has arity $0$, this is a vacuous truth, so any such $s$ constitutes a well-formed formula.
Definition:Matrix Theory
'''Matrix theory''' is the field of mathematics which studies matrices.
Definition:Unit Matrix/Examples/3 x 3
The '''$3 \times 3$ unit matrix''' is as follows: :$\mathbf I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$
Definition:Matrix/Lower Triangular Elements
Let $\mathbf A = \sqbrk a_{m n}$ be a matrix. The elements $a_{i j}: i > j$ are called the '''lower triangular elements'''.
Definition:Matrix/Upper Triangular Elements
Let $\mathbf A = \sqbrk a_{m n}$ be a matrix. The elements $a_{i j}: i < j$ are called the '''upper triangular elements'''.
Definition:Permutation Matrix
A '''permutation matrix (of order $n$)''' is an $n \times n$ square matrix with: : exactly one instance of $1$ in each row and column : $0$ elsewhere.
Definition:Monomial Matrix
A '''monomial matrix (of order $n$)''' is an $n \times n$ square matrix with: :exactly one element in each row and column which is not $0$ :$0$ elsewhere.
Definition:(0, 1)-Matrix
A '''$\tuple {0, 1}$-matrix''' is a matrix whose elements consist only of instances of $0$ and $1$.
Definition:Submatrix/Notation
A submatrix of $\mathbf A$ is denoted as follows. Let: : $\left\{ {a_1, a_2, \ldots, a_r}\right\}$ be the indices of the $r$ selected rows : $\left\{ {b_1, b_2, \ldots, b_s}\right\}$ be the indices of the $s$ selected columns where all of $a_1, \ldots, a_r$ are between $1$ and $m$, and all of $b_1, \ldots, b_s$ are between $1$ and $n$. Then the '''submatrix''' formed from rows $\left\{ {a_1, a_2, \ldots, a_r}\right\}$ and columns $\left\{ {b_1, b_2, \ldots, b_s}\right\}$ is denoted as: :$\mathbf A \left[{a_1, a_2, \ldots, a_r; b_1, b_2, \ldots, b_s}\right]$ It is usual to specify the rows and columns in ascending numerical order.
Definition:Submatrix/Notation/Order (m-1) x (n-1) Submatrix
Let a submatrix $\mathbf B$ of $\mathbf A$ be of order $\left({m - 1}\right) \times \left({n - 1}\right)$. Then it is usual to denote $\mathbf B$ by indicating the (single) row and column of $\mathbf A$ which has been '''removed''', as follows: Let: : $a_j$ be the row of $\mathbf A$ which is not included in $\mathbf B$ : $b_k$ be the column of $\mathbf A$ which is not included in $\mathbf B$. Then the '''submatrix''' $\mathbf B$ formed from the remaining rows and columns of $\mathbf A$ can be denoted as: :$\mathbf A \left({a_j; b_k}\right)$
Definition:Determinant/Matrix/Order
The '''order''' of a determinant is defined as the order of the square matrix on which it is defined.
Definition:Minor of Determinant/Notation/Order n-1
Let a submatrix $\mathbf B$ of $\mathbf A$ be of order $n - 1$. Let: :$j$ be the row of $\mathbf A$ which is not included in $\mathbf B$ :$k$ be the column of $\mathbf A$ which is not included in $\mathbf B$. Thus, let $\mathbf B := \map {\mathbf A} {j; k}$. Then $\map \det {\mathbf B}$ can be denoted: :$D_{i j}$ That is, $D_{i j}$ is the minor of order $n - 1$ obtained from $D$ by deleting all the elements of row $i$ and column $j$.
Definition:Minor of Determinant/Notation
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $D := \map \det {\mathbf A}$ denote the determinant of $\mathbf A$. Let: :$\set {a_1, a_2, \ldots, a_k}$ be the indices of the $k$ selected rows of $\mathbf A$ :$\set {b_1, b_2, \ldots, b_k}$ be the indices of the $k$ selected columns of $\mathbf A$ where all of $a_1, \ldots, a_k$ and all of $b_1, \ldots, b_k$ are between $1$ and $n$. Let: :$\mathbf B := \mathbf A \sqbrk {a_1, a_2, \ldots, a_k; b_1, b_2, \ldots, b_k}$ be the '''submatrix''' formed from rows $\set {a_1, a_2, \ldots, a_k}$ and columns $\set {b_1, b_2, \ldots, b_k}$. The '''order-$k$ minor''' of $D$ formed from rows $r_1, r_2, \ldots, r_k$ and columns $s_1, s_2, \ldots, s_k$ can be denoted: :$\map D {a_1, a_2, \ldots, a_k \mid b_1, b_2, \ldots, b_k}$. Each element of $D$ is an order $1$ minor of $D$, and can be denoted: :$\map D {a_i \mid b_j}$
Definition:Cofactor/Element
Let $a_{r s}$ be an element of $D$. Let $D_{r s}$ be the determinant of order $n-1$ obtained from $D$ by deleting row $r$ and column $s$. Then the '''cofactor''' $A_{r s}$ of the element $a_{r s}$ is defined as: :$A_{r s} := \paren {-1}^{r + s} D_{r s}$
Definition:Cofactor/Minor
Let $D \left({r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}\right)$ be a order-$k$ minor of $D$. Then the '''cofactor''' of $D \left({r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}\right)$ can be denoted: :$\tilde D \left({r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}\right)$ and is defined as: :$\tilde D \left({r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}\right) = \left({-1}\right)^t D \left({r_{k+1}, r_{k+2}, \ldots, r_n \mid s_{k+1}, s_{k+2}, \ldots, s_n}\right)$ where: : $t = r_1 + r_2 + \ldots + r_k + s_1 + s_2 + \ldots s_k$ : $r_{k+1}, r_{k+2}, \ldots, r_n$ are the numbers in $1, 2, \ldots, n$ not in $\left\{{r_1, r_2, \ldots, r_k}\right\}$ : $s_{k+1}, s_{k+2}, \ldots, s_n$ are the numbers in $1, 2, \ldots, n$ not in $\left\{{s_1, s_2, \ldots, s_k}\right\}$ That is, the '''cofactor of a minor''' is the determinant formed from the rows and columns not in that minor, multiplied by the appropriate sign. When $k = 1$, this reduces to the cofactor of an element (as above). When $k = n$, the "minor" is in fact the whole determinant. For convenience its '''cofactor''' is defined as being $1$. Note that the '''cofactor''' of the '''cofactor''' of a minor is the minor itself (multiplied by the appropriate sign).
Definition:Non-Invertible Matrix/Definition 1
Let $\mathbf A$ have no inverse. Then $\mathbf A$ is referred to as '''non-invertible'''.
Definition:Non-Invertible Matrix/Definition 2
Let the determinant of $\mathbf A$ be equal to $0$. Then $\mathbf A$ is referred to as '''non-invertible'''.
Definition:Zero Vector/Euclidean Space
Let $\struct {\R^n, +, \times}_\R$ be a real vector space. The '''zero vector''' in $\struct {\R^n, +, \times}_\R$ is: :$\mathbf 0_{n \times 1} := \begin {bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end {bmatrix}$ where $0 \in \R$.
Definition:Congruence (Number Theory)/Integers/Remainder after Division
'''Congruence modulo $m$''' is defined as the relation $\equiv \pmod m$ on the set of all $a, b \in \Z$: :$a \equiv b \pmod m := \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$ That is, such that $a$ and $b$ have the same remainder when divided by $m$.
Definition:Congruence (Number Theory)/Remainder after Division
We define a relation $\mathcal R_z$ on the set of all $x, y \in \R$: :$\mathcal R_z := \left\{{\left({x, y}\right) \in \R \times \R: \exists k \in \Z: x = y + k z}\right\}$ This relation is called '''congruence modulo $z$''', and the real number $z$ is called the '''modulus'''. When $\left({x, y}\right) \in \mathcal R_z$, we write: :$x \equiv y \pmod z$ and say: : '''$x$ is congruent to $y$ modulo $z$'''.
Definition:Congruence (Number Theory)/Historical Note
The concept of '''congruence modulo an integer''' was first explored by {{AuthorRef|Carl Friedrich Gauss}}. He originated the notation $a \equiv b \pmod m$ in his work {{BookLink|Disquisitiones Arithmeticae|Carl Friedrich Gauss}}, published in $1801$.
Definition:Equivalence Class/Notation
The notation used to denote an equivalence class varies throughout the literature, but is often some variant on the square bracket motif $\eqclass x \RR$. Other variants: * {{BookReference|Lectures in Abstract Algebra|1951|Nathan Jacobson|volume = I|subtitle = Basic Concepts}} uses $\overline x$ for $\eqclass x \RR$. * {{BookReference|Introduction to Topology|1962|Bert Mendelson}} uses $\map \pi x$ for $\eqclass x \RR$. * {{BookReference|Sets and Groups|1965|J.A. Green}} uses $E_x$ for $\eqclass x \RR$. * {{BookReference|Modern Algebra|1965|Seth Warner}} uses $\bigsqcup_\RR \mkern {-28 mu} {\raise 1pt x} \ \ $ for $\eqclass x \RR$, which is even more challenging to render in our installed version of $\LaTeX$ than $\eqclass x \RR$ itself. * {{BookReference|A Handbook of Terms used in Algebra and Analysis|1972|A.G. Howson}} uses $\map {p_\RR} x$. * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth}} uses $x / \RR$ for $\eqclass x \RR$ (compare the notation for quotient set). This source also suggests $x_\RR$ as a variant.
Definition:Equivalence Class/Representative
Let $\eqclass x \RR$ be the equivalence class of $x$ under $\RR$. Let $y \in \eqclass x \RR$. Then $y$ is a '''representative of $\eqclass x \RR$.
Definition:Word (Natural Language)
A '''word''' in natural language is intuitively understood as a sequence of sounds which expresses a concept. When written down, it appears as a sequence of letters, each one of which either is, or contributes to, a [https://en.wikipedia.org/wiki/Phoneme phoneme].
Definition:Scope of Occurrence
Let $\FF$ be a formal language with alphabet $\AA$. Let $\mathbf A$ be a well-formed formula of $\FF$. Let $a$ be a fixed occurrence of some element of $\AA$ in $\mathbf A$. Then the '''scope''' of $a$ is the smallest well-formed part of $\mathbf A$ that contains $a$.
Definition:Universal Closure of Well-Formed Formula
Let $\mathcal L_1$ be the language of predicate logic. Let $\mathbf A$ be a well-formed formula of $\mathcal L_1$. A '''universal closure''' of $\mathbf A$ is a sentence $\mathbf B$ of $\mathcal L_1$ of the form: :$\forall x_1: \cdots \forall x_n: \mathbf A$ By definition of sentence, this means that at least the variables occurring freely in $\mathbf A$ should be quantified over.
Definition:Conjunction/Truth Table/Number
The truth table number of the conjunction operator $p \land q$ is as follows: Ascending order: :$0001$ or $\F \F \F \T$ Descending order: :$1000$ or $\T \F \F \F$
Definition:Truth Table/Number
The '''truth table number''' of the truth table of a statement form is a string consisting of the truth values in numerical form of the main connective. The order in which the '''truth table number''' is presented is usually in ascending or descending numerical order of the truth values of the simple statements contributing to that statement form, either: :Ascending order: $00, 01, 10, 11$, that is: $FF, FT, TF, TT$ or: :Descending order: $11, 10, 01, 00$, that is: $TT, TF, FT, FF$ Which format is being used needs to be specified. Hence the '''truth table number''' can be found by reading either '''up''' or '''down''' the column holding the main connective in the truth table of the statement form.
Definition:Conditional/Truth Table/Number
The truth table number of the conjunction operator $p \land q$ is as follows: Ascending order: :$1101$ or $\T \T \F \T$ Descending order: :$1011$ or $\T \F \T \T$
Definition:Disjunction/Truth Table/Number
The truth table number of the conjunction operator $p \lor q$ is as follows: Ascending order: :$0111$ or $\F \T \T \T$ Descending order: :$1110$ or $\T \T \T \F$
Definition:Biconditional/Truth Table/Number
The truth table number of the biconditional operator $p \iff q$ is as follows: Ascending and descending order: :$1001$ or $\T \F \F \T$
Definition:Exclusive Or/Truth Table/Number
The truth table number of the exclusive or operator $p \oplus q$ is as follows: Ascending and descending order: :$0110$ or $\F \T \T \F$
Definition:Structure for Predicate Logic
Let $\LL_1$ be the language of predicate logic. A '''structure $\AA$ for $\LL_1$''' comprises: :$(1): \quad$ A non-empty set $A$; :$(2): \quad$ For each function symbol $f$ of arity $n$, a mapping $f_\AA: A^n \to A$; :$(3): \quad$ For each predicate symbol $p$ of arity $n$, a mapping $p_\AA: A^n \to \Bbb B$ where $\Bbb B$ denotes the set of truth values. $A$ is called the '''underlying set''' of $\AA$. $f_\AA$ and $p_\AA$ are called the '''interpretations''' of $f$ and $p$ in $\AA$, respectively.
Definition:Assignment for Structure/Formula
Let $\mathbf A$ be a well-formed formula of $\mathcal L_1$. Denote with $V \left({\mathbf A}\right)$ the variables which occur freely in $\mathbf A$. An '''assignment for $\mathbf A$ in $\mathcal A$''' is a mapping $\sigma$ with codomain $A$, whose domain is subject to the following condition: :$V \left({\mathbf A}\right) \subseteq \operatorname{dom} \left({\sigma}\right) \subseteq \mathrm{VAR}$ That is, the domain of $\sigma$ contains only variables, and at least those with a free occurrence in $\mathbf A$.
Definition:Assignment for Structure/Term
Let $\tau$ be a term of $\mathcal L_1$. Denote with $V \left({\tau}\right)$ the variables which occur in $\tau$. An '''assignment for $\tau$ in $\mathcal A$''' is a mapping $\sigma$ with codomain $A$, whose domain is subject to the following condition: :$V \left({\tau}\right) \subseteq \operatorname{dom} \left({\sigma}\right) \subseteq \mathrm{VAR}$ That is, the domain of $\sigma$ contains only variables, and at least those which occur in $\tau$.
Definition:Language of Predicate Logic/Formal Grammar/Term
The '''terms''' of $\LL_1$ are identified by the following bottom-up grammar: {{begin-axiom}} {{axiom | n = \mathbf T \ \textrm {VAR}         | t = Any variable of $\LL_1$ is a '''term'''; }} {{axiom | n = \mathbf T \ \FF_n         | t = Given an $n$-ary function symbol $f \in \FF_n$ and '''terms''' $\tau_1, \ldots, \tau_n$: :$\map f {\tau_1, \ldots, \tau_n}$ is also a '''term'''. }} {{end-axiom}}
Definition:Local Minimum in Set of Reals
Let $X$ be a subset of $\R$, the set of all real numbers. Let $x$ be a real number, $x \in \R$. $x$ is '''local minimum in set''' $X$  :$x \in X \land \exists y \in \R: y < x \land \left({y \,.\,.\, x}\right) \cap X = \varnothing$
Definition:Variable/Restricted
A '''restricted variable''' is a variable whose values are confined to some only of those of which it is capable.
Definition:Variable/Unrestricted
An '''unrestricted variable''' is a variable whose values are not confined in any way to some only of those of which it is capable.
Definition:Prime Number/Definition 6
Let $p \in \N$ be an integer such that $p \ne 0$ and $p \ne \pm 1$. Then $p$ is a '''prime number'''  :$\forall a, b \in \Z: p \divides a b \implies p \divides a$ or $p \divides b$ where $\divides$ means '''is a divisor of'''.
Definition:Twin Primes
'''Twin primes''' are pairs of prime numbers whose difference is $2$.
Definition:Quadratic Residue/Non-Residue
If there is no such integer $x$ such that $x^2 \equiv a \pmod p$, then $a$ is a '''quadratic non-residue of $p$'''.
Definition:Quadratic Residue/Character
$a$ is either a quadratic residue or a quadratic non-residue of $p$. Whether it is or not is known as the '''quadratic character of $a$ modulo $p$'''.
Definition:Heptagon
A '''heptagon''' is a polygon with exactly $7$ sides. :300px === Regular Heptagon === {{:Definition:Heptagon/Regular}}
Definition:Octagon
An '''octagon''' is a polygon with exactly $8$ sides. :300px === Regular Octagon === {{:Definition:Octagon/Regular}}
Definition:Nonagon
A '''nonagon''' is a polygon with exactly $9$ sides. :300px === Regular Nonagon === {{:Definition:Nonagon/Regular}}
Definition:Hendecagon
A '''hendecagon''' is a polygon with exactly $11$ sides. :300px === Regular Hendecagon === {{:Definition:Hendecagon/Regular}}
Definition:Dodecagon
A '''dodecagon''' is a polygon with exactly $12$ sides. :300px === Regular Dodecagon === {{:Definition:Dodecagon/Regular}}
Definition:Heptadecagon
A '''heptadecagon''' is a polygon with exactly $17$ sides. :300px === Regular Heptadecagon === {{:Definition:Heptadecagon/Regular}}
Definition:Pentagon/Regular
A '''regular pentagon''' is a pentagon which is both equilateral and equiangular. That is, a regular polygon with $5$ sides. That is, a pentagon in which all the sides are the same length, and all the vertices have the same angle: :300px
Definition:Hexagon/Regular
A '''regular hexagon''' is a hexagon which is both equilateral and equiangular. That is, a regular polygon with $6$ sides. That is, a hexagon in which all the sides are the same length, and all the vertices have the same angle: :300px
Definition:Heptagon/Regular
A '''regular heptagon''' is a heptagon which is both equilateral and equiangular. That is, a regular polygon with $7$ sides. That is, a heptagon in which all the sides are the same length, and all the vertices have the same angle: :300px
Definition:Octagon/Regular
A '''regular octagon''' is an octagon which is both equilateral and equiangular. That is, a regular polygon with $8$ sides. That is, an octagon in which all the sides are the same length, and all the vertices have the same angle: :300px
Definition:Nonagon/Regular
A '''regular nonagon''' is a nonagon which is both equilateral and equiangular. That is, a regular polygon with $9$ sides. That is, a nonagon in which all the sides are the same length, and all the vertices have the same angle: :300px
Definition:Decagon/Regular
A '''regular decagon''' is a decagon which is both equilateral and equiangular. That is, a regular polygon with $10$ sides. That is, a decagon in which all the sides are the same length, and all the vertices have the same angle: :300px
Definition:Hendecagon/Regular
A '''regular hendecagon''' is a hendecagon which is both equilateral and equiangular. That is, a regular polygon with $11$ sides. That is, a hendecagon in which all the sides are the same length, and all the vertices have the same angle: :300px
Definition:Heptadecagon/Regular
A '''regular heptadecagon''' is a heptadecagon which is both equilateral and equiangular. That is, a regular polygon with $17$ sides. That is, a heptadecagon in which all the sides are the same length, and all the vertices have the same angle: :300px
Definition:Dodecagon/Regular
A '''regular dodecagon''' is a dodecagon which is both equilateral and equiangular. That is, a regular polygon with $12$ sides. That is, a dodecagon in which all the sides are the same length, and all the vertices have the same angle: :300px