Editing Square (section)

===Coordinates and equations===
[[File:Square equation plot.svg|thumb|right|<math>|x| + |y| = 2</math> plotted on ''[[Cartesian coordinates]]''.]]
A [[unit square]] is a square of side length one. Often it is represented in [[Cartesian coordinate]]s as the square enclosing the points <math>(x,y)</math> that have <math>0\le x\le 1</math> and <math>0\le y\le 1</math>. Its vertices are the four points that have 0 or 1 in each of their coordinates.<ref>{{cite book
 | last1 = Rosenthal | first1 = Daniel
 | last2 = Rosenthal | first2 = David
 | last3 = Rosenthal | first3 = Peter
 | doi = 10.1007/978-3-030-00632-7
 | edition = 2nd
 | isbn = 9783030006327
 | series = Undergraduate Texts in Mathematics
 | page = 108
 | publisher = Springer International Publishing
 | title = A Readable Introduction to Real Mathematics
 | url = https://books.google.com/books?id=JQGQDwAAQBAJ&pg=PA108
 | year = 2018}}</ref>

An axis-parallel square with its center at the point <math>(x_c,y_c)</math> and sides of length <math>2r</math> (where <math>r</math> is the inradius, half the side length) has vertices at the four points <math>(x_c\pm r,y_c\pm r)</math>. Its interior consists of the points <math>(x,y)</math> with <math>\max(|x-x_c|,|y-y_c|) < r</math>, and its boundary consists of the points with <math>\max(|x-x_c|,|y-y_c|)=r</math>.<ref name=iobst>{{cite journal
 | last = Iobst | first = Christopher Simon
 | issue = 1
 | journal = Ohio Journal of School Mathematics
 | pages = 27–31
 | title = Shapes and Their Equations: Experimentation with Desmos
 | date = 14 June 2018
 | url = https://ojs.library.osu.edu/index.php/OJSM/article/view/6367
 | volume = 79}}</ref>

A diagonal square with its center at the point <math>(x_c,y_c)</math> and diagonal of length <math>2R</math> (where <math>R</math> is the circumradius, half the diagonal) has vertices at the four points <math>(x_c\pm R,y_c)</math> and <math>(x_c,y_c\pm R)</math>. Its interior consists of the points <math>(x,y)</math> with <math>|x-x_c|+|y-y_c|<R</math>, and its boundary consists of the points with <math>|x-x_c|+|y-y_c|=R</math>.<ref name=iobst/> For instance the illustration shows a diagonal square centered at the origin <math>(0,0)</math> with circumradius 2, given by the equation <math>|x|+|y|=2</math>.

[[File:A square with Gaussian integer vertices.png|thumb|A square formed by multiplying the complex number {{mvar|p}} by powers of {{mvar|i}}, and its translation obtained by adding another complex number {{mvar|c}}. The background grid shows the [[Gaussian integer]]s.]]
In the [[Complex plane|plane of complex numbers]], multiplication by the [[imaginary unit]] <math>i</math> rotates the other term in the product by 90° around the origin (the number zero). Therefore, if any nonzero complex number <math>p</math> is repeatedly multiplied by <math>i</math>, giving the four numbers <math>p</math>, <math>ip</math>, <math>-p</math>, and <math>-ip</math>, these numbers will form the vertices of a square centered at the origin.<ref>{{citation
 | last = Vince | first = John
 | doi = 10.1007/978-0-85729-154-7
 | isbn = 9780857291547
 | page = 11
 | publisher = Springer | location = London
 | title = Rotation Transforms for Computer Graphics
 | year = 2011| bibcode = 2011rtfc.book.....V
 }}</ref> If one interprets the [[real part]] and [[imaginary part]] of these four complex numbers as Cartesian coordinates, with <math>p=x+iy</math>, then these four numbers have the coordinates <math>(x,y)</math>, <math>(-y,x)</math>, <math>(-x,-y)</math>, and <math>(-y,-x)</math>.<ref>{{cite book
 | last = Nahin | first = Paul
 | isbn = 9781400833894
 | page = [https://books.google.com/books?id=OPyPwaElDvUC&pg=PA54 54]
 | publisher = Princeton University Press
 | title = An Imaginary Tale: The Story of <math>\sqrt{-1}</math>
 | year = 2010}}</ref> This square can be translated to have any other complex number <math>c</math> is center, using the fact that the [[translation (geometry)|translation]] from the origin to <math>c</math> is represented in complex number arithmetic as addition with <math>c</math>.<ref name=numberverse/> The [[Gaussian integer]]s, complex numbers with integer real and imaginary parts, form a [[square lattice]] in the complex plane.<ref name=numberverse>{{cite book
 | last1 = McLeman | first1 = Cam
 | last2 = McNicholas | first2 = Erin
 | last3 = Starr | first3 = Colin
 | doi = 10.1007/978-3-030-98931-6
 | isbn = 9783030989316
 | series = Undergraduate Texts in Mathematics
 | page = [https://books.google.com/books?id=G7OiEAAAQBAJ&pg=PA7 7]
 | publisher = Springer International Publishing
 | title = Explorations in Number Theory: Commuting through the Numberverse
 | year = 2022}}</ref>