Visual Insight
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Mathematics Made VisibleThu, 15 Sep 2016 05:42:54 +0000en-UShourly1https://wordpress.org/?v=4.6.1Togliatti Quintic Surface
http://blogs.ams.org/visualinsight/2016/09/15/togliatti-quintic-surface/
http://blogs.ams.org/visualinsight/2016/09/15/togliatti-quintic-surface/#respondThu, 15 Sep 2016 01:00:55 +0000http://blogs.ams.org/visualinsight/?p=2833quintic surface is one defined by a polynomial equation of degree 5. A nodal surface is one whose only singularities are ordinary double points: that is, points where it looks like the origin of the cone in 3-dimensional space defined by \(x^2 + y^2 = z^2\). A Togliatti surface is a quintic nodal surface with the largest possible number of ordinary double points, namely 31. Here Abdelaziz Nait Merzouk has drawn the real points of a Togliatti surface.]]>http://blogs.ams.org/visualinsight/2016/09/15/togliatti-quintic-surface/feed/0Kummer’s Quartic Surface
http://blogs.ams.org/visualinsight/2016/09/01/kummers-quartic-surface/
http://blogs.ams.org/visualinsight/2016/09/01/kummers-quartic-surface/#respondThu, 01 Sep 2016 01:00:46 +0000http://blogs.ams.org/visualinsight/?p=2813quartic surface is one defined by a polynomial equation of degree 4. An ordinary double point is a point where a surface looks like the origin of the cone in 3-dimensional space defined by $x^2 + y^2 = z^2$. The Kummer surfaces are the quartic surfaces with the largest possible number of ordinary double points, namely 16. This picture by Abdelaziz Nait Merzouk shows the real points of a Kummer surface.]]>http://blogs.ams.org/visualinsight/2016/09/01/kummers-quartic-surface/feed/0Cayley’s Nodal Cubic Surface
http://blogs.ams.org/visualinsight/2016/08/15/cayleys-nodal-cubic-surface/
http://blogs.ams.org/visualinsight/2016/08/15/cayleys-nodal-cubic-surface/#respondMon, 15 Aug 2016 01:00:49 +0000http://blogs.ams.org/visualinsight/?p=2791cubic surface is one defined by a polynomial equation of degree 3. Cayley's nodal cubic surface, drawn above by Abdelaziz Nait Merzouk, is the cubic surface with the largest possible number of ordinary double points and no other singularities: that is, points where it looks like the origin of the cone in 3-dimensional space defined by \(x^2 + y^2 = z^2\). It has 4 ordinary double points, shown here at the vertices of a regular tetrahedron.]]>http://blogs.ams.org/visualinsight/2016/08/15/cayleys-nodal-cubic-surface/feed/0Endrass Octic
http://blogs.ams.org/visualinsight/2016/08/01/endrass-octic/
http://blogs.ams.org/visualinsight/2016/08/01/endrass-octic/#respondMon, 01 Aug 2016 01:00:26 +0000http://blogs.ams.org/visualinsight/?p=2745octic surface is one defined by a polynomial equation of degree 8. The Endrass octic, drawn above by Abdelaziz Nait Merzouk, is currently the octic surface with the largest known number of ordinary double points: that is, points where it looks like the origin of the cone in 3-dimensional space defined by \(x^2 + y^2 = z^2\). It has 168 ordinary double points, while the best known upper bound for a octic surface that's smooth except for such singularities is 174.]]>http://blogs.ams.org/visualinsight/2016/08/01/endrass-octic/feed/0Labs Septic
http://blogs.ams.org/visualinsight/2016/07/15/labs-septic/
http://blogs.ams.org/visualinsight/2016/07/15/labs-septic/#respondFri, 15 Jul 2016 01:00:28 +0000http://blogs.ams.org/visualinsight/?p=2722septic surface is one defined by a polynomial equation of degree 7. The Labs septic, drawn above by Abdelaziz Nait Merzouk, is a septic surface with the maximum possible number of ordinary double points: that is, points where it looks like the origin of the cone in 3-dimensional space defined by \( x^2 + y^2 = z^2\).]]>http://blogs.ams.org/visualinsight/2016/07/15/labs-septic/feed/0Barth Decic
http://blogs.ams.org/visualinsight/2016/07/01/barth-decic/
http://blogs.ams.org/visualinsight/2016/07/01/barth-decic/#respondFri, 01 Jul 2016 01:00:25 +0000http://blogs.ams.org/visualinsight/?p=2666decic surface is one defined by a polynomial equation of degree 6. The Barth decic, drawn here by Abdelaziz Nait Merzouk, is the decic surface with the maximum possible number of ordinary double points: that is, points where it looks like the origin of the cone in 3-dimensional space defined by \(x^2 + y^2 = z^2 \).]]>http://blogs.ams.org/visualinsight/2016/07/01/barth-decic/feed/0Small Stellated Dodecahedron
http://blogs.ams.org/visualinsight/2016/06/15/small-stellated-dodecahedron/
http://blogs.ams.org/visualinsight/2016/06/15/small-stellated-dodecahedron/#commentsWed, 15 Jun 2016 01:00:57 +0000http://blogs.ams.org/visualinsight/?p=2566small stellated dodecahedron, drawn here using Robert Webb's Stella software, is made of 12 pentagrams, or 5-pointed stars, with 5 pentagrams meeting at each vertex. ]]>http://blogs.ams.org/visualinsight/2016/06/15/small-stellated-dodecahedron/feed/8Discriminant of Restricted Quintic
http://blogs.ams.org/visualinsight/2016/06/01/discriminant-of-restricted-quintic/
http://blogs.ams.org/visualinsight/2016/06/01/discriminant-of-restricted-quintic/#commentsWed, 01 Jun 2016 01:00:38 +0000http://blogs.ams.org/visualinsight/?p=2522Greg Egan shows the set of points \((a,b,c)\) for which the quintic \(x^5 + ax^4 + bx^2 + c \) has repeated roots. The plane \(c = 0 \) has been removed. This surface is connected to involutes of a cubical parabola and the discriminant of the icosahedral group.]]>http://blogs.ams.org/visualinsight/2016/06/01/discriminant-of-restricted-quintic/feed/5Discriminant of the Icosahedral Group
http://blogs.ams.org/visualinsight/2016/05/15/discriminant-of-the-icosahedral-group/
http://blogs.ams.org/visualinsight/2016/05/15/discriminant-of-the-icosahedral-group/#respondSun, 15 May 2016 01:00:26 +0000http://blogs.ams.org/visualinsight/?p=2534Greg Egan, shows the 'discriminant' of the symmetry group of the icosahedron. This group acts as linear transformations of \(\mathbb{R}^3\) and thus also \(\mathbb{C}^3\). By a theorem of Chevalley, the space of orbits of this group action is again isomorphic to \(\mathbb{C}^3\). Each point in the surface shown here corresponds to a 'nongeneric' orbit: an orbit with fewer than the maximal number of points. More precisely, the space of nongeneric orbits forms a complex surface in \(\mathbb{C}^3\), called the discriminant, whose intersection with \(\mathbb{R}^3\) is shown here.
]]>http://blogs.ams.org/visualinsight/2016/05/15/discriminant-of-the-icosahedral-group/feed/0Involutes of a Cubical Parabola
http://blogs.ams.org/visualinsight/2016/05/01/involutes-of-a-cubical-parabola/
http://blogs.ams.org/visualinsight/2016/05/01/involutes-of-a-cubical-parabola/#respondSun, 01 May 2016 01:00:26 +0000http://blogs.ams.org/visualinsight/?p=2469Marshall Hampton shows the involutes of the curve \(y = x^3\). It lies at a fascinating mathematical crossroads, which we shall explore in a series of three posts.]]>http://blogs.ams.org/visualinsight/2016/05/01/involutes-of-a-cubical-parabola/feed/0