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class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Commutative diagram</span> </div> </a> <ul id="toc-Commutative_diagram-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hasse_diagrams" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hasse_diagrams"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Hasse diagrams</span> </div> </a> <ul id="toc-Hasse_diagrams-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Knot_diagrams" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Knot_diagrams"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Knot diagrams</span> </div> </a> <ul id="toc-Knot_diagrams-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Venn_diagram" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Venn_diagram"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Venn diagram</span> </div> </a> <ul id="toc-Venn_diagram-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voronoi_diagram" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Voronoi_diagram"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.7</span> <span>Voronoi diagram</span> </div> </a> <ul id="toc-Voronoi_diagram-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wallpaper_group_diagrams" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wallpaper_group_diagrams"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.8</span> <span>Wallpaper group diagrams</span> </div> </a> <ul id="toc-Wallpaper_group_diagrams-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Young_diagram" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Young_diagram"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.9</span> <span>Young diagram</span> </div> </a> <ul id="toc-Young_diagram-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_mathematical_diagrams" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_mathematical_diagrams"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.10</span> <span>Other mathematical diagrams</span> </div> </a> <ul id="toc-Other_mathematical_diagrams-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div 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class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Euclid_Lueneburg_ms_page_8.jpg/420px-Euclid_Lueneburg_ms_page_8.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Euclid_Lueneburg_ms_page_8.jpg/560px-Euclid_Lueneburg_ms_page_8.jpg 2x" data-file-width="848" data-file-height="1227" /></a><figcaption><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a>, ms. from Lüneburg, A.D. 1200</figcaption></figure> <p><b>Mathematical diagrams</b>, such as <a href="/wiki/Chart" title="Chart">charts</a> and <a href="/wiki/Graph_of_a_function" title="Graph of a function">graphs</a>, are mainly designed to convey mathematical relationships—for example, comparisons over time.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Specific_types_of_mathematical_diagrams">Specific types of mathematical diagrams</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_diagram&action=edit&section=1" title="Edit section: Specific types of mathematical diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Argand_diagram">Argand diagram</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_diagram&action=edit&section=2" title="Edit section: Argand diagram"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="skin-invert-image" typeof="mw:File/Thumb"><a href="/wiki/File:Complex_number_illustration.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Complex_number_illustration.svg/120px-Complex_number_illustration.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Complex_number_illustration.svg/180px-Complex_number_illustration.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Complex_number_illustration.svg/240px-Complex_number_illustration.svg.png 2x" data-file-width="180" data-file-height="180" /></a><figcaption>Argand diagram.</figcaption></figure> <p>A <a href="/wiki/Complex_number" title="Complex number">complex number</a> can be visually represented as a pair of numbers forming a vector on a diagram called an <a href="/wiki/Argand_diagram" class="mw-redirect" title="Argand diagram">Argand diagram</a> The <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> is sometimes called the <i>Argand plane</i> because it is used in <i>Argand diagrams</i>. These are named after <a href="/wiki/Jean-Robert_Argand" title="Jean-Robert Argand">Jean-Robert Argand</a> (1768–1822), although they were first described by Norwegian-Danish land surveyor and mathematician <a href="/wiki/Caspar_Wessel" title="Caspar Wessel">Caspar Wessel</a> (1745–1818).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Argand diagrams are frequently used to plot the positions of the <a href="/wiki/Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">poles</a> and <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">zeroes</a> of a <a href="/wiki/Mathematical_function" class="mw-redirect" title="Mathematical function">function</a> in the complex plane. </p><p>The concept of the complex plane allows a <a href="/wiki/Geometric" class="mw-redirect" title="Geometric">geometric</a> interpretation of complex numbers. Under <a href="/wiki/Addition" title="Addition">addition</a>, they add like <a href="/wiki/Vector_(spatial)" class="mw-redirect" title="Vector (spatial)">vectors</a>. The <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> of two complex numbers can be expressed most easily in <a href="/wiki/Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</a> — the magnitude or <i>modulus</i> of the product is the product of the two <a href="/wiki/Absolute_value" title="Absolute value">absolute values</a>, or moduli, and the angle or <i>argument</i> of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation. </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Butterfly-FFT.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Butterfly-FFT.png/120px-Butterfly-FFT.png" decoding="async" width="120" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Butterfly-FFT.png/180px-Butterfly-FFT.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/98/Butterfly-FFT.png/240px-Butterfly-FFT.png 2x" data-file-width="250" data-file-height="315" /></a><figcaption>Butterfly diagram</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Butterfly_diagram">Butterfly diagram</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_diagram&action=edit&section=3" title="Edit section: Butterfly diagram"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the context of <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> algorithms, a <a href="/wiki/Butterfly_diagram" title="Butterfly diagram">butterfly</a> is a portion of the computation that combines the results of smaller <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transforms</a> (DFTs) into a larger DFT, or vice versa (breaking a larger DFT up into subtransforms). The name "butterfly" comes from the shape of the data-flow diagram in the radix-2 case, as described below. The same structure can also be found in the <a href="/wiki/Viterbi_algorithm" title="Viterbi algorithm">Viterbi algorithm</a>, used for finding the most likely sequence of hidden states. </p><p>The <a href="/wiki/Butterfly_diagram" title="Butterfly diagram">butterfly diagram</a> show a data-flow diagram connecting the inputs <i>x</i> (left) to the outputs <i>y</i> that depend on them (right) for a "butterfly" step of a radix-2 <a href="/wiki/Cooley%E2%80%93Tukey_FFT_algorithm" title="Cooley–Tukey FFT algorithm">Cooley–Tukey FFT algorithm</a>. This diagram resembles a <a href="/wiki/Butterfly" title="Butterfly">butterfly</a> as in the <a href="/wiki/Morpho_(genus)" title="Morpho (genus)">Morpho butterfly</a> shown for comparison, hence the name. </p> <figure class="skin-invert-image" typeof="mw:File/Thumb"><a href="/wiki/File:5_lemma.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/5_lemma.svg/300px-5_lemma.svg.png" decoding="async" width="300" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/5_lemma.svg/450px-5_lemma.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/5_lemma.svg/600px-5_lemma.svg.png 2x" data-file-width="388" data-file-height="131" /></a><figcaption>A commutative diagram depicting the <a href="/wiki/Five_lemma" title="Five lemma">five lemma</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Commutative_diagram">Commutative diagram</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_diagram&action=edit&section=4" title="Edit section: Commutative diagram"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Commutative_diagram" title="Commutative diagram">Commutative diagram</a></div> <p>In mathematics, and especially in <a href="/wiki/Category_theory" title="Category theory">category theory</a>, a commutative diagram is a diagram of <a href="/wiki/Object_(category_theory)" class="mw-redirect" title="Object (category theory)">objects</a>, also known as vertices, and <a href="/wiki/Morphism" title="Morphism">morphisms</a>, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition. </p><p>Commutative diagrams play the role in category theory that equations play in algebra. </p> <figure class="mw-halign-left skin-invert-image" typeof="mw:File/Thumb"><a href="/wiki/File:Hasse_diagram_of_powerset_of_3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Hasse_diagram_of_powerset_of_3.svg/120px-Hasse_diagram_of_powerset_of_3.svg.png" decoding="async" width="120" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Hasse_diagram_of_powerset_of_3.svg/180px-Hasse_diagram_of_powerset_of_3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Hasse_diagram_of_powerset_of_3.svg/240px-Hasse_diagram_of_powerset_of_3.svg.png 2x" data-file-width="429" data-file-height="325" /></a><figcaption>Hasse diagram.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Hasse_diagrams">Hasse diagrams</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_diagram&action=edit&section=5" title="Edit section: Hasse diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a> is a simple picture of a finite <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</a>, forming a <a href="/wiki/Graph_drawing" title="Graph drawing">drawing</a> of the partial order's <a href="/wiki/Transitive_reduction" title="Transitive reduction">transitive reduction</a>. Concretely, one represents each element of the set as a vertex on the page and draws a line segment or curve that goes upward from <i>x</i> to <i>y</i> precisely when <i>x</i> < <i>y</i> and there is no <i>z</i> such that <i>x</i> < <i>z</i> < <i>y</i>. In this case, we say y <a href="/wiki/Covering_relation" title="Covering relation">covers</a> x, or y is an immediate successor of x. In a Hasse diagram, it is required that the curves be drawn so that each meets exactly two vertices: its two endpoints. Any such diagram (given that the vertices are labeled) uniquely determines a partial order, and any partial order has a unique transitive reduction, but there are many possible placements of elements in the plane, resulting in different Hasse diagrams for a given order that may have widely varying appearances. </p> <figure class="mw-halign-right skin-invert-image" typeof="mw:File/Thumb"><a href="/wiki/File:TrefoilKnot_01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/TrefoilKnot_01.svg/120px-TrefoilKnot_01.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/TrefoilKnot_01.svg/180px-TrefoilKnot_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/04/TrefoilKnot_01.svg/240px-TrefoilKnot_01.svg.png 2x" data-file-width="250" data-file-height="250" /></a><figcaption>Knot diagram.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Knot_diagrams">Knot diagrams</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_diagram&action=edit&section=6" title="Edit section: Knot diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Knot_theory" title="Knot theory">Knot theory</a> a useful way to visualise and manipulate knots is to project the knot onto a plane—;think of the knot casting a shadow on the wall. A small perturbation in the choice of projection will ensure that it is <a href="/wiki/Injective_function" title="Injective function">one-to-one</a> except at the double points, called <i>crossings</i>, where the "shadow" of the knot crosses itself once transversely<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>At each crossing we must indicate which section is "over" and which is "under", so as to be able to recreate the original knot. This is often done by creating a break in the strand going underneath. If by following the diagram the knot alternately crosses itself "over" and "under", then the diagram represents a particularly well-studied class of knot, <a href="/wiki/Alternating_knot" title="Alternating knot">alternating knots</a>. </p> <figure class="mw-halign-right skin-invert-image" typeof="mw:File/Thumb"><a href="/wiki/File:Venn_diagram_cmyk.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Venn_diagram_cmyk.svg/120px-Venn_diagram_cmyk.svg.png" decoding="async" width="120" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Venn_diagram_cmyk.svg/180px-Venn_diagram_cmyk.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Venn_diagram_cmyk.svg/240px-Venn_diagram_cmyk.svg.png 2x" data-file-width="471" data-file-height="450" /></a><figcaption>Venn diagram.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Venn_diagram">Venn diagram</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_diagram&action=edit&section=7" title="Edit section: Venn diagram"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a> is a representation of mathematical sets: a mathematical diagram representing sets as circles, with their relationships to each other expressed through their overlapping positions, so that all possible relationships between the sets are shown.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The Venn diagram is constructed with a collection of simple closed curves drawn in the plane. The principle of these diagrams is that classes be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram. That is, the diagram initially leaves room for any possible relation of the classes, and the actual or given relation, can then be specified by indicating that some particular region is null or is not null.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Voronoi_centerlines_skeleton.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Voronoi_centerlines_skeleton.gif/120px-Voronoi_centerlines_skeleton.gif" decoding="async" width="120" height="72" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Voronoi_centerlines_skeleton.gif/180px-Voronoi_centerlines_skeleton.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/92/Voronoi_centerlines_skeleton.gif/240px-Voronoi_centerlines_skeleton.gif 2x" data-file-width="1071" data-file-height="640" /></a><figcaption>Voronoi centerlines.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Voronoi_diagram">Voronoi diagram</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_diagram&action=edit&section=8" title="Edit section: Voronoi diagram"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Voronoi_diagram" title="Voronoi diagram">Voronoi diagram</a> is a special kind of decomposition of a <a href="/wiki/Metric_space" title="Metric space">metric space</a> determined by distances to a specified discrete set of objects in the space, e.g., by a <a href="/wiki/Discrete_set" class="mw-redirect" title="Discrete set">discrete set</a> of points. This diagram is named after <a href="/wiki/Georgy_Voronoi" class="mw-redirect" title="Georgy Voronoi">Georgy Voronoi</a>, also called a Voronoi <a href="/wiki/Tessellation" title="Tessellation">tessellation</a>, a Voronoi decomposition, or a Dirichlet tessellation after <a href="/wiki/Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Peter Gustav Lejeune Dirichlet</a>. </p><p>In the simplest case, we are given a set of points S in the plane, which are the Voronoi sites. Each site s has a Voronoi cell V(s) consisting of all points closer to s than to any other site. The segments of the Voronoi diagram are all the points in the plane that are equidistant to two sites. The Voronoi nodes are the points equidistant to three (or more) sites </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Wallpaper_group_diagram_p4m_square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group_diagram_p4m_square.svg/120px-Wallpaper_group_diagram_p4m_square.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group_diagram_p4m_square.svg/180px-Wallpaper_group_diagram_p4m_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Wallpaper_group_diagram_p4m_square.svg/240px-Wallpaper_group_diagram_p4m_square.svg.png 2x" data-file-width="425" data-file-height="425" /></a><figcaption>Wallpaper group diagram.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Wallpaper_group_diagrams">Wallpaper group diagrams</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_diagram&action=edit&section=9" title="Edit section: Wallpaper group diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Wallpaper_group" title="Wallpaper group">wallpaper group</a> or <i>plane symmetry group</i> or <i>plane crystallographic group</i> is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. There are 17 possible distinct <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>. </p><p>Wallpaper groups are two-dimensional <a href="/wiki/Symmetry_groups" class="mw-redirect" title="Symmetry groups">symmetry groups</a>, intermediate in complexity between the simpler <a href="/wiki/Frieze_group" title="Frieze group">frieze groups</a> and the three-dimensional <a href="/wiki/Crystallographic_group" class="mw-redirect" title="Crystallographic group">crystallographic groups</a>, also called <a href="/wiki/Space_group" title="Space group">space groups</a>. Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns which are very different in style, color, scale or orientation may belong to the same group. </p> <div class="mw-heading mw-heading3"><h3 id="Young_diagram">Young diagram</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_diagram&action=edit&section=10" title="Edit section: Young diagram"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i>Young diagram</i> or <a href="/wiki/Young_tableau" title="Young tableau">Young tableau</a>, also called <a href="/wiki/Ferrers_diagram" class="mw-redirect" title="Ferrers diagram">Ferrers diagram</a>, is a finite collection of boxes, or cells, arranged in left-justified rows, with the row sizes weakly decreasing (each row has the same or shorter length than its predecessor). </p> <figure class="mw-halign-left skin-invert-image" typeof="mw:File/Thumb"><a href="/wiki/File:Young_diagram_for_541_partition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Young_diagram_for_541_partition.svg/120px-Young_diagram_for_541_partition.svg.png" decoding="async" width="120" height="72" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Young_diagram_for_541_partition.svg/180px-Young_diagram_for_541_partition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Young_diagram_for_541_partition.svg/240px-Young_diagram_for_541_partition.svg.png 2x" data-file-width="180" data-file-height="108" /></a><figcaption>Young diagram.</figcaption></figure> <p>Listing the number of boxes in each row gives a <a href="/wiki/Partition_(number_theory)" class="mw-redirect" title="Partition (number theory)">partition</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /></span> of a positive integer <i>n</i>, the total number of boxes of the diagram. The Young diagram is said to be of shape <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /></span>, and it carries the same information as that partition. Listing the number of boxes in each column gives another partition, the <b>conjugate</b> or <i>transpose</i> partition of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /></span>; one obtains a Young diagram of that shape by reflecting the original diagram along its main diagonal. </p><p>Young tableaux were introduced by <a href="/wiki/Alfred_Young_(mathematician)" title="Alfred Young (mathematician)">Alfred Young</a>, a <a href="/wiki/Mathematician" title="Mathematician">mathematician</a> at <a href="/wiki/University_of_Cambridge" title="University of Cambridge">Cambridge University</a>, in 1900. They were then applied to the study of symmetric group by <a href="/wiki/Georg_Frobenius" class="mw-redirect" title="Georg Frobenius">Georg Frobenius</a> in 1903. Their theory was further developed by many mathematicians. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Other_mathematical_diagrams">Other mathematical diagrams</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_diagram&action=edit&section=11" title="Edit section: Other mathematical diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Cremona_diagram" title="Cremona diagram">Cremona diagram</a></li> <li><a href="/wiki/De_Finetti_diagram" title="De Finetti diagram">De Finetti diagram</a></li> <li><a href="/wiki/Dynkin_diagram" title="Dynkin diagram">Dynkin diagram</a></li> <li><a href="/wiki/Elementary_diagram" title="Elementary diagram">Elementary diagram</a></li> <li><a href="/wiki/Euler_diagram" title="Euler diagram">Euler diagram</a></li> <li><a href="/wiki/Stellation_diagram" title="Stellation diagram">Stellation diagram</a></li> <li><a href="/wiki/Ulam_spiral" title="Ulam spiral">Ulam spiral</a></li> <li><a href="/wiki/Van_Kampen_diagram" title="Van Kampen diagram">Van Kampen diagram</a></li> <li><a href="/wiki/Taylor_diagram" title="Taylor diagram">Taylor diagram</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_diagram&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Logic_diagram" class="mw-redirect" title="Logic diagram">Logic diagram</a></li> <li><a href="/wiki/Mathematical_jargon" class="mw-redirect" title="Mathematical jargon">Mathematical jargon</a></li> <li><a href="/wiki/Mathematical_model" title="Mathematical model">Mathematical model</a></li> <li><a href="/wiki/Mathematics_as_a_language" class="mw-redirect" title="Mathematics as a language">Mathematics as a language</a></li> <li><a href="/wiki/Mathematical_visualization" title="Mathematical visualization">Mathematical visualization</a></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_diagram&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://openlearn.open.ac.uk/course/view.php?id=2993&topic=all">Working with diagrams</a> at LearningSpace.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Wessel's memoir was presented to the Danish Academy in 1797; Argand's paper was published in 1806.<br />(<style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWhittakerWatson1927" class="citation book cs1">Whittaker, Edmund Taylor; Watson, G.N. (1927). <i><a href="/wiki/A_Course_of_Modern_Analysis" title="A Course of Modern Analysis">A Course of Modern Analysis</a>: An Introduction to the General Theory of Infinite Processes and of Analytic Functions, with an Account of the Principal Transcendental Functions</i>. Cambridge University Press. p. 9. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-58807-2" title="Special:BookSources/978-0-521-58807-2"><bdi>978-0-521-58807-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+of+Modern+Analysis%3A+An+Introduction+to+the+General+Theory+of+Infinite+Processes+and+of+Analytic+Functions%2C+with+an+Account+of+the+Principal+Transcendental+Functions&rft.pages=9&rft.pub=Cambridge+University+Press&rft.date=1927&rft.isbn=978-0-521-58807-2&rft.aulast=Whittaker&rft.aufirst=Edmund+Taylor&rft.au=Watson%2C+G.N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+diagram" class="Z3988"></span>)</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRolfsen1976" class="citation book cs1">Rolfsen, Dale (1976). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qFLvAAAAMAAJ"><i>Knots and links</i></a>. Publish or Perish. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-914098-16-4" title="Special:BookSources/978-0-914098-16-4"><bdi>978-0-914098-16-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Knots+and+links&rft.pub=Publish+or+Perish&rft.date=1976&rft.isbn=978-0-914098-16-4&rft.aulast=Rolfsen&rft.aufirst=Dale&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqFLvAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+diagram" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://encarta.msn.com/dictionary_1861733754/venn_diagram.html">"Venn diagram"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091107154423/http://encarta.msn.com/dictionary_1861733754/Venn_diagram.html">Archived</a> 2009-11-07 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Encarta World English Dictionary, North American Edition 2007. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091107154423/http://encarta.msn.com/dictionary_1861733754/Venn_diagram.html">Archived</a> 2009-11-01.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="/wiki/Clarence_Irving_Lewis" class="mw-redirect" title="Clarence Irving Lewis">Clarence Irving Lewis</a> (1918). <a rel="nofollow" class="external text" href="https://archive.org/details/asurveyofsymboli00lewiuoft"><i>A Survey of Symbolic Logic</i></a>. Republished in part by Dover in 1960. p. 157.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_diagram&action=edit&section=14" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBarker-PlummerBailin1997" class="citation journal cs1">Barker-Plummer, Dave; Bailin, Sidney C. (1997). "The Role of Diagrams in Mathematical Proofs". <i>Machine Graphics and Vision</i>. <b>6</b> (1): <span class="nowrap">25–</span>56. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.49.4712">10.1.1.49.4712</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Machine+Graphics+and+Vision&rft.atitle=The+Role+of+Diagrams+in+Mathematical+Proofs&rft.volume=6&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E25-%3C%2Fspan%3E56&rft.date=1997&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.49.4712%23id-name%3DCiteSeerX&rft.aulast=Barker-Plummer&rft.aufirst=Dave&rft.au=Bailin%2C+Sidney+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+diagram" class="Z3988"></span> (Special Issue on Diagrammatic Representation and Reasoning).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBarker-PlummerBailin2001" class="citation book cs1">Barker-Plummer, Dave; Bailin, Sidney C. (2001). "On the practical semantics of mathematical diagrams". In Anderson, M. (ed.). <i>Reasoning with Diagrammatic Representations</i>. <a href="/wiki/Springer_Verlag" class="mw-redirect" title="Springer Verlag">Springer Verlag</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.9246">10.1.1.30.9246</a></span>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-85233-242-6" title="Special:BookSources/978-1-85233-242-6"><bdi>978-1-85233-242-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+the+practical+semantics+of+mathematical+diagrams&rft.btitle=Reasoning+with+Diagrammatic+Representations&rft.pub=Springer+Verlag&rft.date=2001&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.30.9246%23id-name%3DCiteSeerX&rft.isbn=978-1-85233-242-6&rft.aulast=Barker-Plummer&rft.aufirst=Dave&rft.au=Bailin%2C+Sidney+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKidman,_G.2002" class="citation book cs1">Kidman, G. (2002). "The Accuracy of mathematical diagrams in curriculum materials". In Cockburn, A.; Nardi, E. (eds.). <i>Proceedings of the PME 26</i>. Vol. 3. University of East Anglia. pp. <span class="nowrap">201–</span>8.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Accuracy+of+mathematical+diagrams+in+curriculum+materials&rft.btitle=Proceedings+of+the+PME+26&rft.pages=%3Cspan+class%3D%22nowrap%22%3E201-%3C%2Fspan%3E8&rft.pub=University+of+East+Anglia&rft.date=2002&rft.au=Kidman%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKulpa,_Zenon2004" class="citation book cs1"><a href="/wiki/Zenon_Kulpa" title="Zenon Kulpa">Kulpa, Zenon</a> (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=j8bdlF8-2sIC&pg=PA191">"On Diagrammatic Representation of Mathematical Knowledge"</a>. In Andréa Asperti; Bancerek, Grzegorz; Trybulec, Andrzej (eds.). <i>Mathematical knowledge management: third international conference, MKM 2004, Białowieża, Poland, September 19–21, 2004 : Proceedings</i>. Springer. pp. <span class="nowrap">191–</span>204. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-23029-8" title="Special:BookSources/978-3-540-23029-8"><bdi>978-3-540-23029-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+Diagrammatic+Representation+of+Mathematical+Knowledge&rft.btitle=Mathematical+knowledge+management%3A+third+international+conference%2C+MKM+2004%2C+Bia%C5%82owie%C5%BCa%2C+Poland%2C+September+19%E2%80%9321%2C+2004+%3A+Proceedings&rft.pages=%3Cspan+class%3D%22nowrap%22%3E191-%3C%2Fspan%3E204&rft.pub=Springer&rft.date=2004&rft.isbn=978-3-540-23029-8&rft.au=Kulpa%2C+Zenon&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dj8bdlF8-2sIC%26pg%3DPA191&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPuphaiboon,_K.Woodcock,_A.Scrivener,_S.2005" class="citation book cs1">Puphaiboon, K.; Woodcock, A.; Scrivener, S. (25 March 2005). "Design method for developing mathematical diagrams". In Bust, Philip D.; McCabe, P.T. (eds.). <i>Contemporary ergonomics 2005 Proceedings of the International Conference on Contemporary Ergonomics (CE2005)</i>. Taylor & Francis. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-415-37448-4" title="Special:BookSources/978-0-415-37448-4"><bdi>978-0-415-37448-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Design+method+for+developing+mathematical+diagrams&rft.btitle=Contemporary+ergonomics+2005+Proceedings+of+the+International+Conference+on+Contemporary+Ergonomics+%28CE2005%29&rft.pub=Taylor+%26+Francis&rft.date=2005-03-25&rft.isbn=978-0-415-37448-4&rft.au=Puphaiboon%2C+K.&rft.au=Woodcock%2C+A.&rft.au=Scrivener%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+diagram" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_diagram&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Mathematical_diagram" class="extiw" title="commons:Mathematical diagram"><span style="font-style:italic; font-weight:bold;">Mathematical diagram</span></a>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="http://plato.stanford.edu/archives/fall2008/entries/diagrams/">"Diagrams"</a>. <i>The Stanford Encyclopedia of Philosophy</i>. Fall 2008.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Diagrams&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.date=2008&rft_id=http%3A%2F%2Fplato.stanford.edu%2Farchives%2Ffall2008%2Fentries%2Fdiagrams%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKulpa,_Zenon" class="citation web cs1"><a href="/wiki/Zenon_Kulpa" title="Zenon Kulpa">Kulpa, Zenon</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130425225258/http://www.ippt.gov.pl/~zkulpa/diagrams/">"Diagrammatics: The art of thinking with diagrams"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.ippt.gov.pl/~zkulpa/diagrams/">the original</a> on April 25, 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Diagrammatics%3A+The+art+of+thinking+with+diagrams&rft.au=Kulpa%2C+Zenon&rft_id=http%3A%2F%2Fwww.ippt.gov.pl%2F~zkulpa%2Fdiagrams%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+diagram" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cass/euclid/papyrus/papyrus.html">One of the oldest extant diagrams from Euclid</a> by Otto Neugebauer</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLomas,_Dennis1998" class="citation web cs1">Lomas, Dennis (1998). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110721195349/http://www.ed.uiuc.edu/eps/PES-Yearbook/1998/lomas.html">"Diagrams in Mathematical Education: A Philosophical Appraisal"</a>. Philosophy of Education Society. Archived from <a rel="nofollow" class="external text" href="http://www.ed.uiuc.edu/eps/PES-Yearbook/1998/lomas.html">the original</a> on 2011-07-21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Diagrams+in+Mathematical+Education%3A+A+Philosophical+Appraisal&rft.pub=Philosophy+of+Education+Society&rft.date=1998&rft.au=Lomas%2C+Dennis&rft_id=http%3A%2F%2Fwww.ed.uiuc.edu%2Feps%2FPES-Yearbook%2F1998%2Flomas.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+diagram" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output 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href="/wiki/Gaspard_Monge" title="Gaspard Monge">Gaspard Monge</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">19th century</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Charles_Dupin" title="Charles Dupin">Charles Dupin</a></li> <li><a href="/wiki/Adolphe_Quetelet" title="Adolphe Quetelet">Adolphe Quetelet</a></li> <li><a href="/wiki/Andr%C3%A9-Michel_Guerry" title="André-Michel Guerry">André-Michel Guerry</a></li> <li><a href="/wiki/William_Playfair" title="William Playfair">William Playfair</a></li> <li><a href="/wiki/August_Kekul%C3%A9" title="August Kekulé">August Kekulé</a></li> <li><a href="/wiki/Charles_Joseph_Minard" title="Charles Joseph Minard">Charles Joseph Minard</a></li> <li><a href="/wiki/Francis_Amasa_Walker" title="Francis Amasa Walker">Francis Amasa Walker</a></li> <li><a href="/wiki/John_Venn" title="John Venn">John Venn</a></li> <li><a href="/wiki/Oliver_Byrne_(mathematician)" title="Oliver Byrne (mathematician)">Oliver Byrne</a></li> <li><a href="/wiki/Matthew_Henry_Phineas_Riall_Sankey" title="Matthew Henry Phineas Riall Sankey">Matthew Sankey</a></li> <li><a href="/wiki/Charles_Booth_(social_reformer)" title="Charles Booth (social reformer)">Charles Booth</a></li> <li><a href="/wiki/John_Snow" title="John Snow">John Snow</a></li> <li><a href="/wiki/Florence_Nightingale" title="Florence Nightingale">Florence Nightingale</a></li> <li><a href="/wiki/Karl_Wilhelm_Pohlke" title="Karl Wilhelm Pohlke">Karl Wilhelm Pohlke</a></li> <li><a href="/wiki/Toussaint_Loua" title="Toussaint Loua">Toussaint Loua</a></li> <li><a href="/wiki/Francis_Galton" title="Francis Galton">Francis Galton</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Early 20th century</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Edward_Walter_Maunder" title="Edward Walter Maunder">Edward Walter Maunder</a></li> <li><a href="/wiki/Otto_Neurath" title="Otto Neurath">Otto Neurath</a></li> <li><a href="/wiki/W._E._B._Du_Bois" title="W. E. B. Du Bois">W. E. B. Du Bois</a></li> <li><a href="/wiki/Henry_Gantt" title="Henry Gantt">Henry Gantt</a></li> <li><a href="/wiki/Arthur_Lyon_Bowley" title="Arthur Lyon Bowley">Arthur Lyon Bowley</a></li> <li><a href="/wiki/Howard_G._Funkhouser" title="Howard G. Funkhouser">Howard G. Funkhouser</a></li> <li><a href="/wiki/John_B._Peddle" title="John B. Peddle">John B. Peddle</a></li> <li><a href="/wiki/Ejnar_Hertzsprung" title="Ejnar Hertzsprung">Ejnar Hertzsprung</a></li> <li><a href="/wiki/Henry_Norris_Russell" title="Henry Norris Russell">Henry Norris Russell</a></li> <li><a href="/wiki/Max_O._Lorenz" title="Max O. Lorenz">Max O. Lorenz</a></li> <li><a href="/wiki/Fritz_Kahn" title="Fritz Kahn">Fritz Kahn</a></li> <li><a href="/wiki/Harry_Beck" title="Harry Beck">Harry Beck</a></li> <li><a href="/wiki/Erwin_Raisz" title="Erwin Raisz">Erwin Raisz</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mid 20th century</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Jacques_Bertin" title="Jacques Bertin">Jacques Bertin</a></li> <li><a href="/wiki/Rudolf_Modley" title="Rudolf Modley">Rudolf Modley</a></li> <li><a href="/wiki/Arthur_H._Robinson" title="Arthur H. Robinson">Arthur H. Robinson</a></li> <li><a href="/wiki/John_Tukey" title="John Tukey">John Tukey</a></li> <li><a href="/wiki/Mary_Eleanor_Spear" title="Mary Eleanor Spear">Mary Eleanor Spear</a></li> <li><a href="/wiki/Edgar_Anderson" title="Edgar Anderson">Edgar Anderson</a></li> <li><a href="/wiki/Howard_T._Fisher" title="Howard T. Fisher">Howard T. Fisher</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Late 20th century</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Borden_Dent" title="Borden Dent">Borden Dent</a></li> <li><a href="/wiki/Nigel_Holmes" title="Nigel Holmes">Nigel Holmes</a></li> <li><a href="/wiki/William_S._Cleveland" title="William S. Cleveland">William S. Cleveland</a></li> <li><a href="/wiki/George_G._Robertson" title="George G. Robertson">George G. Robertson</a></li> <li><a href="/wiki/Bruce_H._McCormick" title="Bruce H. McCormick">Bruce H. McCormick</a></li> <li><a href="/wiki/Catherine_Plaisant" title="Catherine Plaisant">Catherine Plaisant</a></li> <li><a href="/wiki/Stuart_Card" title="Stuart Card">Stuart Card</a></li> <li><a href="/wiki/Pat_Hanrahan" title="Pat Hanrahan">Pat Hanrahan</a></li> <li><a href="/wiki/Edward_Tufte" title="Edward Tufte">Edward Tufte</a></li> <li><a href="/wiki/Ben_Shneiderman" title="Ben Shneiderman">Ben Shneiderman</a></li> <li><a href="/wiki/Michael_Friendly" title="Michael Friendly">Michael Friendly</a></li> <li><a href="/wiki/Howard_Wainer" title="Howard Wainer">Howard Wainer</a></li> <li><a href="/wiki/Clifford_A._Pickover" title="Clifford A. Pickover">Clifford A. Pickover</a></li> <li><a href="/wiki/Lawrence_J._Rosenblum" title="Lawrence J. Rosenblum">Lawrence J. Rosenblum</a></li> <li><a href="/wiki/Thomas_A._DeFanti" title="Thomas A. DeFanti">Thomas A. DeFanti</a></li> <li><a href="/wiki/George_Furnas" title="George Furnas">George Furnas</a></li> <li><a href="/wiki/Sheelagh_Carpendale" title="Sheelagh Carpendale">Sheelagh Carpendale</a></li> <li><a href="/wiki/Cynthia_Brewer" title="Cynthia Brewer">Cynthia Brewer</a></li> <li><a href="/wiki/Jock_D._Mackinlay" title="Jock D. Mackinlay">Jock D. Mackinlay</a></li> <li><a href="/wiki/Alan_MacEachren" title="Alan MacEachren">Alan MacEachren</a></li> <li><a href="/wiki/David_Goodsell" title="David Goodsell">David Goodsell</a></li> <li><a href="/wiki/Kwan-Liu_Ma" title="Kwan-Liu Ma">Kwan-Liu Ma</a></li> <li><a href="/wiki/Michael_Maltz" class="mw-redirect" title="Michael Maltz">Michael Maltz</a></li> <li><a href="/wiki/Leland_Wilkinson" title="Leland Wilkinson">Leland Wilkinson</a></li> <li><a href="/wiki/Alfred_Inselberg" title="Alfred Inselberg">Alfred Inselberg</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Early 21st century</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ben_Fry" title="Ben Fry">Ben Fry</a></li> <li><a href="/wiki/Jeffrey_Heer" title="Jeffrey Heer">Jeffrey Heer</a></li> <li><a href="/wiki/Jessica_Hullman" title="Jessica Hullman">Jessica Hullman</a></li> <li><a href="/wiki/Gordon_Kindlmann" title="Gordon Kindlmann">Gordon Kindlmann</a></li> <li><a href="/wiki/Aaron_Koblin" title="Aaron Koblin">Aaron Koblin</a></li> <li><a href="/wiki/Christopher_R._Johnson" title="Christopher R. Johnson">Christopher R. Johnson</a></li> <li><a href="/wiki/Manuel_Lima" title="Manuel Lima">Manuel Lima</a></li> <li><a href="/wiki/David_McCandless" title="David McCandless">David McCandless</a></li> <li><a href="/wiki/Mauro_Martino" title="Mauro Martino">Mauro Martino</a></li> <li><a href="/wiki/John_Maeda" title="John Maeda">John Maeda</a></li> <li><a href="/wiki/Miriah_Meyer" title="Miriah Meyer">Miriah Meyer</a></li> <li><a href="/wiki/Tamara_Munzner" title="Tamara Munzner">Tamara Munzner</a></li> <li><a href="/wiki/Ade_Olufeko" class="mw-redirect" title="Ade Olufeko">Ade Olufeko</a></li> <li><a href="/wiki/Hanspeter_Pfister" title="Hanspeter Pfister">Hanspeter Pfister</a></li> <li><a href="/wiki/Hans_Rosling" title="Hans Rosling">Hans Rosling</a></li> <li><a href="/wiki/Claudio_Silva_(computer_scientist)" title="Claudio Silva (computer scientist)">Claudio Silva</a></li> <li><a href="/wiki/Moritz_Stefaner" title="Moritz Stefaner">Moritz Stefaner</a></li> <li><a href="/wiki/Fernanda_Vi%C3%A9gas" title="Fernanda Viégas">Fernanda Viégas</a></li> <li><a href="/wiki/Martin_M._Wattenberg" title="Martin M. Wattenberg">Martin Wattenberg</a></li> <li><a href="/wiki/Bang_Wong" title="Bang Wong">Bang Wong</a></li> <li><a href="/wiki/Hadley_Wickham" title="Hadley Wickham">Hadley Wickham</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related <br />topics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cartography" title="Cartography">Cartography</a></li> <li><a href="/wiki/Chartjunk" title="Chartjunk">Chartjunk</a></li> <li><a href="/wiki/Color_coding_in_data_visualization" class="mw-redirect" title="Color coding in data visualization">Color coding</a></li> <li><a href="/wiki/Computer_graphics" title="Computer graphics">Computer graphics</a> <ul><li><a href="/wiki/Computer_graphics_(computer_science)" title="Computer graphics (computer science)">in computer science</a></li></ul></li> <li><a href="/wiki/CPK_coloring" title="CPK coloring">CPK coloring</a></li> <li><a href="/wiki/Graph_drawing" title="Graph drawing">Graph drawing</a></li> <li><a href="/wiki/Graphic_design" title="Graphic design">Graphic design</a></li> <li><a href="/wiki/Graphic_organizer" title="Graphic organizer">Graphic organizer</a></li> <li><a href="/wiki/Imaging_science" class="mw-redirect" title="Imaging science">Imaging science</a></li> <li><a href="/wiki/Information_art" title="Information art">Information art</a></li> <li><a href="/wiki/Infographic" title="Infographic">Information graphics</a></li> <li><a href="/wiki/Information_science" title="Information science">Information science</a></li> <li><a href="/wiki/Misleading_graph" title="Misleading graph">Misleading graph</a></li> <li><a href="/wiki/Neuroimaging" title="Neuroimaging">Neuroimaging</a></li> <li><a href="/wiki/Patent_drawing" title="Patent drawing">Patent drawing</a></li> <li><a href="/wiki/Scientific_modelling" title="Scientific modelling">Scientific modelling</a></li> <li><a href="/wiki/Spatial_analysis" title="Spatial analysis">Spatial analysis</a></li> <li><a href="/wiki/Visual_analytics" title="Visual analytics">Visual analytics</a></li> <li><a href="/wiki/Visual_perception" title="Visual perception">Visual perception</a></li> <li><a href="/wiki/Volume_cartography" title="Volume cartography">Volume cartography</a></li> <li><a href="/wiki/Volume_rendering" title="Volume rendering">Volume rendering</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Mathematical_diagram&oldid=1278891718">https://en.wikipedia.org/w/index.php?title=Mathematical_diagram&oldid=1278891718</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Diagrams" title="Category:Diagrams">Diagrams</a></li><li><a href="/wiki/Category:Mathematical_concepts" title="Category:Mathematical concepts">Mathematical concepts</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Commons_link_is_on_Wikidata" title="Category:Commons link is on Wikidata">Commons link is on Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 5 March 2025, at 06:43<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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