An octic surface is one defined by a polynomial equation of degree 8. This image by Abdelaziz Nait Merzouk shows an octic discovered by Chmutov with 154 real ordinary double points or nodes: that is, points where it looks like the origin of the cone in 3-dimensional space defined by x2+y2=z2.
The ambidextrous moving sofa problem is to find the planar shape of maximal area that can negotiate right-angled turns both to the right and to the left in a hallway of width 1. The current best known solution was found by Dan Romik, and is shown here.
This is an image of the truncated {6,3,3} honeycomb in hyperbolic space.
The Bunimovich stadium is a rectangle capped by semicircles in which a point particle moves at constant speed along straight lines, reflecting off the boundary in a way that the angle of incidence equals the angle of reflection. This animation, made by Phillipe Roux, shows a collection of such particles initially moving in the same…
A nonic surface is one defined by a polynomial equation of degree 9. This image by Juan García Escudero shows a nonic surface called Q9, which has 220 real ordinary double points: that is, points where it looks like the origin of the cone in 3-dimensional space defined by x2+y2=z2.
This picture by Greg Egan shows the Laves graph, a structure discovered by the crystallographer Fritz Laves in 1932. It is also called the ‘K4 crystal’, since is an embedding of the maximal abelian cover of the complete graph on 4 vertices in 3-dimensional Euclidean space. It is also called the ‘triamond’, since it is a theoretically possible — but never yet seen — crystal structure for carbon.
This picture by Greg Egan shows the pattern of carbon atoms in a diamond, called the diamond cubic. Each atom is bonded to four neighbors. This pattern is found not just in carbon but also other elements in the same column of the periodic table: silicon, germanium, and tin.
A quintic 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 x2+y2=z2. 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.
A quartic 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 x2+y2=z2. 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.
A cubic 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 x2+y2=z2. It has 4 ordinary double points, shown here at the vertices of a regular tetrahedron.
An octic 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 x2+y2=z2. 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.
A septic 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 x2+y2=z2.
A decic 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 x2+y2=z2.
The small 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.
This image by Greg Egan shows the set of points (a,b,c) for which the quintic x5+ax4+bx2+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.
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