Name: Projections of the Higman-Sims graph from the Leech lattice @davidamadore
Uploaded: 2026-09-26T08:17:00-05:00
Duration: 8 min 17 s
Description: Projections of the Higman-Sims graph from the Leech lattice

Projections of the Higman-Sims graph from the Leech lattice

David Madore 2026-09-26 | Projections of the Higman-Sims graph from the Leech lattice

Rotating transparent icosahedron

David Madore 2022-08-31 | A regular icosahedron with index of refraction of 2.4 (roughly that of diamond) being slowly rotated around one of its axes of symmetry of order 5. Made with PoV-Ray. PoV-Ray source: gist.github.com/Gro-Tsen/e97998801b6e5fb25db426dc757b585a (change ior from 1.5 to 2.4)

Evolution of Julia sets as parameter dives into crevice with argument 9/31

David Madore 2020-03-04 | Evolution of Julia sets as their parameter point follows the parameter ray with external argument 9/31, diving into the crevice between the main cardioid and its argument 2/5 bud. The position of the parameter point is shown in red in the Mandelbrot inset at the bottom left. Its external argument is constantly 9/31 and it approaches the bud's root (~ −0.481763 + +0.531657i) exponentially slowly.

Evolution of Julia sets as parameter dives into crevice with argument 1/3

David Madore 2020-03-04 | Evolution of Julia sets as their parameter point follows the parameter ray with external argument 1/3, diving into the crevice between the main cardioid and its period 2 bud. The position of the parameter point is shown in red in the Mandelbrot inset at the bottom left. Its external argument is constantly 1/3 and it approaches the bud's root (−¾) exponentially slowly.

Evolution of Julia sets as parameter circles around −¾

David Madore 2020-03-04 | Evolution of Julia sets as their parameter point moves in a circle around the point −¾ which is the root of the period 2 component (i.e., the point where that bud attaches to the main cardioid). The position of the parameter point is shown in red in the Mandelbrot inset at the bottom left.

Evolution of Julia sets as parameter moves into a sub-bulb in the Mandelbrot set

David Madore 2020-02-16 | Evolution of Julia sets as their parameter point moves away from the center of the largest (or period 2) bulb of the Mandelbrot set, leaves it at a rational point (here with rotation angle 2/5 turn), and crosses into the bulb attached there. The position of the parameter point is shown in red in the Mandelbrot inset at the bottom left. It moves away from the center while keeping the same multiplier angle of the 2-cycle.

Evolution of Julia sets as parameter changes component in the Mandelbrot set

David Madore 2020-02-16 | Evolution of Julia sets as their parameter point moves away from the center of the Mandelbrot set, leaves the cardioid at an rational point (here with rotation angle 2/5 turn), and crosses into the bulb attached there. The position of the parameter point is shown in red in the Mandelbrot inset at the bottom left. It moves away from the center while keeping the same multiplier angle of the alpha fixed point.

Evolution of Julia sets as parameter leaves the largest bulb of the Mandelbrot set

David Madore 2020-02-11 | Evolution of Julia sets as their parameter point moves away from the center of the largest (or period 2) bulb of the Mandelbrot set and leaves it at an irrational point (here with rotation angle conjugate to the golden ratio). The position of the parameter point is shown in red in the Mandelbrot inset at the bottom left. It moves away from the center while keeping the same multiplier angle of the 2-cycle.

Evolution of Julia sets as parameter leaves the Mandelbrot set

David Madore 2020-02-11 | Evolution of Julia sets as their parameter point moves away from the center of the Mandelbrot set and leaves the cardioid at an irrational point (here with rotation angle conjugate to the golden ratio). The position of the parameter point is shown in red in the Mandelbrot inset at the bottom left. It moves away from the center while keeping the same multiplier angle of the alpha fixed point.

Evolution of Julia sets as parameter walks the edge of the cardioid of the Mandelbrot set

David Madore 2020-02-10 | Evolution of Julia sets as their parameter point moves about the edge of the cardiod (period 1 component) of the Mandelbrot set (counterclockwise from root to duplication point). The position of the parameter point is shown in red in the Mandelbrot inset at the bottom left. It moves around the cardiod while always staying ever so slightly inside the boundary.

Evolution of Julia sets as parameter walks the edge of the largest bulb of the Mandelbrot set

David Madore 2020-02-10 | Evolution of Julia sets as their parameter point moves about the edge of the largest (or period 2) bulb of the Mandelbrot set (counterclockwise from bulb root to duplication point). The position of the parameter point is shown in red in the Mandelbrot inset at the bottom left. It moves around the bulb while always staying ever so slightly inside the boundary, following a circle.

Evolution of Julia sets as parameter walks around the Mandelbrot set

David Madore 2020-02-10 | Evolution of Julia sets as their parameter point moves around the Mandelbrot set (counterclockwise from cardioid root to antenna tip). The position of the parameter point is shown in red in the Mandelbrot inset at the bottom left. It moves around the Mandelbrot set while always staying outside it, following a contour line (equipotential) and crossing external ray lines at constant speed.

The sound of some grassmannians

David Madore 2019-08-23 | The vibrational mode spectra of some compact Riemannian symmetric spaces — namely, real, complex and quaternionic grassmannians as well as the octonionic projective plane — converted to sound.

See twitter.com/gro_tsen/status/1164864635163545601 for further discussion.

Random waves on a flat torus (triangular lattice)

David Madore 2019-08-09 | This video can be described either as an analogue of youtube.com/watch?v=vod6z379S7g but with random initial data (and three color channels instead of one), or as an analogue of youtube.com/watch?v=T9y22RNj69c but on a flat torus instead of a sphere. See the descriptions of the aforementioned videos for more details.

Growing circles on a flat torus (square lattice)

David Madore 2019-04-23 | Circles of linearly growing radius centered around the points of a square lattice.

Compare with youtube.com/watch?v=vod6z379S7g — the scales and speeds of both videos have been chosen identically, so they can be compared time-for-time.

Growing circles on a flat torus (triangular lattice)

David Madore 2019-04-23 | Circles of linearly growing radius centered around the points of a triangular lattice.

Compare with youtube.com/watch?v=cLVQ67SwNZE — the scales and speeds of both videos have been chosen identically, so they can be compared time-for-time.

Projections of the Higman-Sims graph from the Leech lattice (new version)

David Madore 2018-10-30 | This is a higher resolution version of youtube.com/watch?v=neUd794Gbg0 (the two are not frame-for-frame identical because I lost the exact parameters used in constructing the old video). Sadly, YouTube's aggressive compression makes the quality quite poor anyway.

An even better version is the animated JavaScript page: http://www.madore.org/~david/math/higman-sims-graph.html (note that source code is readable for those who want to know who it's done).

Also see on Wikimedia commons: commons.wikimedia.org/wiki/Category:Higman-Sims_Graph

Waves on a flat torus (square lattice)

David Madore 2018-06-28 | An animation of a solution of the wave equation (that is, (∂²/∂t²−c²Δ)φ=0 where Δ is the Laplacian) on the flat torus ℝ²/L where L is a square lattice. Equivalently, this is the wave equation on the plane applied to an L-periodic function. The initial condition is a fairly peaked Gaussian.

Values are expressed by a color gradient: black is zero, white is positive (red even more so) and blue is negative.

Waves on a flat torus (triangular lattice)

David Madore 2018-06-28 | An animation of a solution of the wave equation (that is, (∂²/∂t²−c²Δ)φ=0 where Δ is the Laplacian) on the flat torus ℝ²/L where L is an equilateral triangular lattice. Equivalently, this is the wave equation on the plane applied to an L-periodic function. The initial condition is a fairly peaked Gaussian.

Values are expressed by a color gradient: black is zero, white is positive (red even more so) and blue is negative.

A JavaScript equivalent of the same computation (lower quality, but can last forever) is available at bjacob.github.io/webgl_waves/torus_waves.html (thanks to Benoit Jacob).

Fourier transform of an icosahedron (Coxeter plane view)

David Madore 2018-04-23 | This video shows the Fourier transform of the vertices of a regular icosahedron (i.e., of a sum of Dirac δ distributions, one at each vertex of the icosahedron) as a 3D function viewed through 2D slices whose direction is a Coxeter plane of the icosahedron.

Positive values are represented as shades of gray and negative values as shades of blue. The origin is in the center of the frame at the middle of the video (24″).

The vertices are at: (0, ±1, ±ϕ), (±1, ±ϕ, 0), (±ϕ, 0, ±1), where ϕ is the golden ratio, and the plane sections are orthogonal to (0,2+ϕ,1+3ϕ).

(The earlier video youtube.com/watch?v=uyitkl00Ey4 shows the same thing but with a different slicing and a different color gradient.)

Fourier transform of a dodecahedron (Coxeter plane view)

David Madore 2018-04-23 | This video shows the Fourier transform of the vertices of a regular dodecahedron (i.e., of a sum of Dirac δ distributions, one at each vertex of the dodecahedron) as a 3D function viewed through 2D slices whose direction is a Coxeter plane of the dodecahedron.

Positive values are represented as shades of gray and negative values as shades of blue. The origin is in the center of the frame at the middle of the video (24″).

The vertices are at: (±1,±1,±1), (0,±ϕ,±(ϕ−1)), (±(ϕ−1),0,±ϕ), (±ϕ,±(ϕ−1),0), where ϕ is the golden ratio, and the plane sections are orthogonal to (0,2+ϕ,1+3ϕ).

(The earlier video youtube.com/watch?v=hmeIWFg6dxU shows the same thing but with a different slicing and a different color gradient.)

Fourier transform of the E8 root system

David Madore 2018-03-15 | This video shows a three-dimensional cross-section of the Fourier transform of the E8 root system (or Gosset 4_21 polytope: see youtube.com/watch?v=E-LC_l3gNuc for a direct projection of this polytope).

The Fourier transform of the E8 root system (or more accurately, of a sum of Dirac δ distributions, one at each root of the system) is the sum of 240 complex exponentials (or 120 cosines), one for each root of E8. This function takes values between −16 and +240 (represented using an ad hoc color scheme, see below), it is periodic modulo the E8 (coroot, i.e., dual) lattice, with values +240 exactly on the points of the latter.

Here we take a three-dimensional cross-section of the (eight-dimensional) space, with two dimensions displayed as image coordinates and the third dimension as time (i.e., each frame is a two-dimensional cross-section, and this section is translated uniformly in time, perpendicularly to its plane, in a random direction). The plane direction in this video has been chosen to be a Coxeter plane for the lattice, which explains the 30-fold symmetry which occurs exactly around a lattice point and approximately in various places.

The section, axes and scale are exactly the same as in the video youtube.com/watch?v=LPVT8aDK2pc so it can be said to show a different view of the same space (the two videos correspond frame per frame and pixel per pixel).

A lattice point (which we can call "origin") is encountered exactly in the middle of the video, at the center of the screen. So at this point, an exact 30-fold symmetry is encountered. (Of course, everything is symmetric around this point.)

The color scheme is a piecewise linear gradient as follows: −16 is bright green, −8 is black, 0 is white, +16 is bright red, and +240 is bright blue (these are some of the critical values of the function).

Sounds of Lie group representations

David Madore 2017-06-25 | A representation (character) of a compact Lie group, restricted to the "Kostant principal SU₂" has a weight spectrum which can be interpreted as a sound. Here we listen to the sound thus produced by various representations of various Lie groups (the frequency is set so that the fundamental representation of SU₂ maps to 110Hz, or the A note of the second octave).

All compact simple Lie groups up to rank 7 have been included, and for each one, every fundamental representation and the adjoint representation.

Fourier transform of an icosahedron

David Madore 2017-04-24 | This video shows the Fourier transform of the vertices of a regular icosahedron (i.e., of a sum of Dirac δ distributions, one at each vertex of the dodecahedron) as a 3D function viewed through 2D slices, represented as levels of gray.

The vertices are at: (0, ±1, ±ϕ), (±1, ±ϕ, 0), (±ϕ, 0, ±1).

Fourier transform of a dodecahedron

David Madore 2017-04-24 | This video shows the Fourier transform of the vertices of a regular dodecahedron (i.e., of a sum of Dirac δ distributions, one at each vertex of the dodecahedron) as a 3D function viewed through 2D slices, represented as levels of gray.

The vertices are at: (±1,±1,±1), (0,±(ϕ−1),±ϕ), (±(ϕ−1),±ϕ,0), (±ϕ,0,±(ϕ−1)), where ϕ is the golden ratio.

Voronoi cells of the E8 lattice: 3D cross section with symmetry of order 24, distance map

David Madore 2017-04-14 | This video is similar to youtube.com/watch?v=LPVT8aDK2pc but with a plane exhibiting symmetry of order 24 instead of 30.

The E8 lattice (or Gosset lattice) is a very regular lattice in 8 dimensions (it is generated by the E8 root system, and realizes the optimal sphere packing in 8 dimensions). The Voronoi cell of a lattice point is the set of points closer to that lattice point than to any other: the Voronoi cells of the E8 lattice are polytopes with 240 facets and 19440 vertices (of two kinds, 2160 corresponding to the "deep" holes of E8 and 17280 to the "shallow" holes), and the Voronoi diagram of E8 is the tiling of 8-space by these polytopes.

This video shows a three-dimensional cross-section of the Voronoi diagram of E8, with two dimensions displayed as image coordinates and the third dimension as time (i.e., each frame is a two-dimensional cross-section, and this section is translated uniformly in time, perpendicularly to its plane, in a random direction).

The plane direction in this video has been chosen to be a symmetry plane of order 24 for the lattice, which explains the 24-fold symmetry which occurs exactly around a lattice point and approximately in various places.

A lattice point (which we can call "origin") is encountered exactly in the middle of the video, at the center of the screen. So at this point, an exact 24-fold symmetry is encountered. (Of course, everything is symmetric around this point.)

Voronoi cells of the E8 lattice: random 3D cross section, distance map

David Madore 2017-04-14 | This video is similar to youtube.com/watch?v=VSAnezNAjLg but what has been plotted here is the (squared) distance to the nearest lattice point (black=0, white=1).

The E8 lattice (or Gosset lattice) is a very regular lattice in 8 dimensions (it is generated by the E8 root system, and realizes the optimal sphere packing in 8 dimensions). The Voronoi cell of a lattice point is the set of points closer to that lattice point than to any other: the Voronoi cells of the E8 lattice are polytopes with 240 facets and 19440 vertices (of two kinds, 2160 corresponding to the "deep" holes of E8 and 17280 to the "shallow" holes), and the Voronoi diagram of E8 is the tiling of 8-space by these polytopes.

This video shows a three-dimensional cross-section of the Voronoi diagram of E8, with two dimensions displayed as image coordinates and the third dimension as time (i.e., each frame is a random two-dimensional cross-section, and this section is translated uniformly in time, perpendicularly to its plane, in a random direction).

Voronoi cells of the E8 lattice: 3D cross section with Coxeter plane, distance map

David Madore 2017-04-14 | This video is similar to youtube.com/watch?v=w7y8HA-k_b4 but what has been plotted here is the (squared) distance to the nearest lattice point (black=0, white=1).

The E8 lattice (or Gosset lattice) is a very regular lattice in 8 dimensions (it is generated by the E8 root system, and realizes the optimal sphere packing in 8 dimensions). The Voronoi cell of a lattice point is the set of points closer to that lattice point than to any other: the Voronoi cells of the E8 lattice are polytopes with 240 facets and 19440 vertices (of two kinds, 2160 corresponding to the "deep" holes of E8 and 17280 to the "shallow" holes), and the Voronoi diagram of E8 is the tiling of 8-space by these polytopes.

This video shows a three-dimensional cross-section of the Voronoi diagram of E8, with two dimensions displayed as image coordinates and the third dimension as time (i.e., each frame is a two-dimensional cross-section, and this section is translated uniformly in time, perpendicularly to its plane, in a random direction).

The plane direction in this video has been chosen to be a Coxeter plane for the lattice, which explains the 30-fold symmetry which occurs exactly around a lattice point and approximately in various places.

A lattice point (which we can call "origin") is encountered exactly in the middle of the video, at the center of the screen. So at this point, an exact 30-fold symmetry is encountered. (Of course, everything is symmetric around this point.)

Voronoi cells of the A8 lattice: 3D cross section with Coexter plane

David Madore 2017-04-11 | This video is similar to youtube.com/watch?v=t7zVizg3LMU but the lattice here is A8 instead of E8. (Its Coxeter plane exhibits order 9 symmetry instead of order 30 for E8.)

Voronoi cells of the D8 lattice: 3D cross section with Coxeter plane

David Madore 2017-04-10 | This video is similar to youtube.com/watch?v=t7zVizg3LMU but the lattice here is D8 instead of E8. (Its Coxeter plane exhibits order 14 symmetry instead of order 30 for E8.)

Voronoi cells of the E8 lattice: 3D cross section with Coxeter plane (wider view)

David Madore 2017-04-10 | See youtube.com/watch?v=w7y8HA-k_b4 for description: this is the same thing but with a view 3× larger in both directions.

Voronoi cells of the E8 lattice: a three-dimensional cross-section with Coxeter plane

David Madore 2017-04-07 | The E8 lattice (or Gosset lattice) is a very regular lattice in 8 dimensions (it is generated by the E8 root system, and realizes the optimal sphere packing in 8 dimensions). The Voronoi cell of a lattice point is the set of points closer to that lattice point than to any other: the Voronoi cells of the E8 lattice are polytopes with 240 facets and 19440 vertices (of two kinds, 2160 corresponding to the "deep" holes of E8 and 17280 to the "shallow" holes), and the Voronoi diagram of E8 is the tiling of 8-space by these polytopes.

This video shows a three-dimensional cross-section of the Voronoi diagram of E8, with two dimensions displayed as image coordinates and the third dimension as time (i.e., each frame is a random two-dimensional cross-section, and this section is translated uniformly in time, perpendicularly to its plane). Each colored region corresponds to a slice of the Voronoi cell of a lattice point. The colors are given by three additional perpendicular directions.

Unlike the companion video youtube.com/watch?v=VSAnezNAjLg (which shows a random cross-section), the plane direction in this video has been chosen to be a Coxeter plane for the lattice, which explains the 30-fold symmetry which occurs exactly around a lattice point and approximately in various places.

A lattice point (which we can call "origin") is encountered exactly in the middle of the video, at the center of the screen. So at this point, an exact 30-fold symmetry is encountered. (Of course, everything is symmetric around this point.)

Voronoi cells of the E8 lattice: a random three-dimensional cross-section

David Madore 2017-04-07 | The E8 lattice (or Gosset lattice) is a very regular lattice in 8 dimensions (it is generated by the E8 root system, and realizes the optimal sphere packing in 8 dimensions). The Voronoi cell of a lattice point is the set of points closer to that lattice point than to any other: the Voronoi cells of the E8 lattice are polytopes with 240 facets and 19440 vertices (of two kinds, 2160 corresponding to the "deep" holes of E8 and 17280 to the "shallow" holes), and the Voronoi diagram of E8 is the tiling of 8-space by these polytopes.

This video shows a random three-dimensional cross-section of the Voronoi diagram of E8, with two dimensions displayed as image coordinates and the third dimension as time (i.e., each frame is a random two-dimensional cross-section, and this section is translated uniformly in time, perpendicularly to its plane). Each colored region corresponds to a slice of the Voronoi cell of a lattice point. The colors are given by three additional perpendicular directions.

A lattice point (which we can call "origin") is encountered exactly in the middle of the video, at the center of the screen. (Of course, everything is symmetric around this point.)

Spherical Waves

David Madore 2015-09-08 | An animation of a solution of the wave equation on the sphere (that is, (∂²/∂t²−c²Δ)φ=0 where Δ is the spherical Laplacian, and the displayed function φ is a 3-component vector giving the RGB channels). The initial condition is somewhat random. More precisely, the function is computed through its decomposition in spherical harmonics φ = ∑u[ℓ,m](t)·Y[ℓ,m] (for −ℓ≤m≤ℓ, and here ℓ only ranges from 0 through 30 because of computational limits), where Y[ℓ,m] are L²-normalized eigenvectors of the Laplacian (ΔY[ℓ,m] = −ℓ(ℓ+1)·Y[ℓ,m]), and u[ℓ,m](t) is a sinusoidal function of t with frequency √(ℓ(ℓ+1)), its phase and amplitude being chosen at random.

An interactive version in JavaScript (but with fewer spherical harmonics) is available at http://www.madore.org/~david/math/spherical-waves-webgl.html (or at http://www.madore.org/~david/math/spherical-waves.html without WebGL).

Deforming a rotation by 2 full turns to a trivial one (sequential version)

David Madore 2015-04-25 | This video illustrates how a rotation of a sphere by two full turns can be deformed continuously into a non-rotation.

This is a sequential version showing the same deformation as in this other version: http://www.youtube.com/watch?v=HK7KksT1nYw

This video consists of 25 segments, each 8 seconds long: in the first segment, the sphere performs a rotation by two full turns (=4π radians) around a constant axis. Each of the following segments modifies the movement a little bit until, in the end, the sphere is not moving at all.

It would not be possible to similarly deform to nothing a rotation of a circle (whatever the number of turns), because the "number of turns" is a well-defined invariant. Perhaps more surprisingly, it would also be impossible to deform to nothing a rotation of the sphere by a single full turn: this is because a single-turn rotation defines a non-trivial element of the spin group, whereas a rotation by two full turns gives the identity (=trivial) element of the spin group. The spin group is simply connected, so any loop therein can be contracted to nothing, and this is what is being done here.

Deforming a rotation by 2 full turns to a trivial one (illustrating the Spin group)

David Madore 2015-04-25 | This video illustrates how a rotation of a sphere by two full turns can be deformed continuously into a non-rotation.

During one period of the video (=8 seconds), the top-left sphere (labeled 0) performs a rotation by two full turns (=4π radians) around a constant axis, while the bottom-right sphere (labeled 27) remains fixed. Each of the intermediate spheres performs a movement starting from the same starting position for each and ending in that same position (8″ later). Each sphere's movement is very close to the previous and next ones, thus illustrating how we can continuously deform the first movement (a rotation by two full turns) to the last (no rotation).

(The video repeats 6 times in the hope of making it easier to understand what is happening. Note that the spheres are labeled by column, the columns being read alternatively top-to-bottom and bottom-to-top.)

A variant of this video, where the various movements are performed sequentially instead of being laid out as a grid, is available as http://www.youtube.com/watch?v=fiatLbvsObs

It would not be possible to similarly deform to nothing a rotation of a circle (whatever the number of turns), because the "number of turns" is a well-defined invariant. Perhaps more surprisingly, it would also be impossible to deform to nothing a rotation of the sphere by a single full turn: this is because a single-turn rotation defines a non-trivial element of the spin group, whereas a rotation by two full turns gives the identity (=trivial) element of the spin group. The spin group is simply connected, so any loop therein can be contracted to nothing, and this is what is being done here.

Deformation of a hexagon with fixed sides and angles

David Madore 2014-06-05 | Every solid segment shown on this video maintains a constant length, yet the figure is able to deform, showing that convexity is essential in Cauchy's theorem on the rigidity of polyhedra. The hexagon formed by the gray segments has equal sides and equal angles (constantly right angles). The red and green triangles are equilateral. The whole figure is reminiscent of cyclohexane (but with different angles).

Accelerating toward the speed of light (combined view)

David Madore 2014-04-01 | A simulation of a uniform acceleration reaching 99.93% of the speed of light in 1′40″, illustrating the Doppler effect, aberration of light and time dilation.

See the description of http://www.youtube.com/watch?v=0uunSMipnxA for a detailed account of what is being represented.

The upper left quadrant represents the front view (identical to the aforementioned video), the upper right quadrant represents the view to the right, the bottom right quadrant represents the rear view, and the bottom left quadrant represents the current coordinates (s = proper time in seconds; t = time as measured by fixed observers, also in seconds; z = distance traveled in light-seconds; dt/ds = γ = Lorentz factor; dz/ds = apparent speed measured as distance traveled in the fixed background over proper time; dz/dt = speed as a fraction of the speed of light).

Accelerating toward the speed of light

David Madore 2014-03-29 | A simulation of a uniform acceleration reaching 99.93% of the speed of light in 1′40″, illustrating the Doppler effect, aberration of light and time dilation.

(See also http://www.youtube.com/watch?v=BYdcCOD8poo for an alternate view of the same thing.)

The observer is moving along a straight line, halfway between two infinite "fixed" tiled planes, and constantly accelerating at 0.04c/s = 11992km/s² (or 1.2 million gees: this is the inertial force felt by the observer).

The "floor" below the observer consists of square tiles, 5 light-seconds in side (1.5 million km or 0.01AU — under 4 times the Earth-Moon distance). It is 1.25 light seconds "under" the observer. The edges of the tiles are perfectly straight lines; they emit light as a black body spectrum at 6500K (approximately the color of the surface of the sun), while the center of the times are as one of 3250K ("brown"; the ratio of their luminosities is also 16, as for the corresponding black bodies).

The "ceiling" above the observer consists of square tiles of the same size (the edges are thinner, but the period is still 5 light seconds). Unlike the floor, they emit monochromatic light, with a wavelength of approximately 635nm (the color of a typical red laser pointer). The edges of the ceiling tiles is twice brighter than the tiles themselves, but of the same color. Furthermore, the tiles blink all simultaneously (with a period of 10s): they all remain on for 5s then off for 5s. (The edges don't blink, only the center part blinks. The tiles are all perfectly synchronized, in the "fixed" reference frame.) The blinking appears as a series of concentric circles simply because the speed of light is finite (and the edge of a tile is equal to the distance that light travels in a blink half-period, i.e., 5s).

Furthermore, space has been made slightly absorptive of light (again with a characteristic length (=optical depth) of 5 light-seconds), simply so as to give a better sense of depth (this is the reason things fade off in the distance: without absorption, a perfectly emitting surface would have equal apparent brightness all the way to infinity).

The following can be observed:

* The Doppler effect causes light coming from ahead to be blueshifted. This is why the ceiling shifts in color. first becoming green then forming a rainbow. The rainbow is a combination of Doppler effect and aberration of light. Inside the rainbow, light from the ceiling is blueshifted too far in the ultraviolet to be visible, outside it is redshifted too far in the infrared. The Doppler effect is less chromatically manifest on the floor because a Doppler shifted black body spectrum is still a black body spectrum, but it is responsible for the glaring white light ahead and everything on the edges becoming dark.

* Aberration of light causes the picture to be contracted toward the center of the field of vision. This is visible before any noticeable movement (=parallax) has been made, which is why the observer may give the impression of moving backwards during the first few seconds. Also, the floor lines which are perpendicular to the observer's motion ("perpendicular" in the fixed reference frame, that is) soon seem to converge to points on the horizon which get nearer and nearer to the point directly ahead. (Remark: aberration of light produces a Möbius transformation (=homography) on the sphere of light directions. So straight lines remain spherical circles — before plane projection that is.)

* The speed of light, of course, can never be reached: even though the moving observer ship is constantly accelerating (as witnessed by accelerometers aboard the ship, measuring the inertial force), fixed observers will see the ship tend toward the speed of light like a hyperbolic tangent. At the end of the video, the speed is 99.93%=tanh(4) of that of light, corresponding to a rapidity of 4 (rapidity increases linearly throughout the video). The distance traveled in the entire sequence is 657.7 light-seconds (or 131.5 tiles crossed, or 1.32AU — where 1AU is the Earth-Sun distance) and fixed observer will measured an elapsed time of 682.2 seconds (average speed: 96.4% of that of light), distinctly longer than the 100 seconds that the moving observer feels (and at the end of the video, the moving observer sees the ceiling blink 27.3 times faster than normal — even though, surprisingly, they would see a fixed clock tick slower). This is how we can travel over six times more than 100 light-seconds in just 100 seconds of proper time.

As we tend to the speed of light, everything we see of the fixed objects tends to just one infinitely bright dot ahead, and darkness all around.

Ideal rotation of the hyperbolic plane

David Madore 2013-12-04 | The effects of an "ideal rotation" of the hyperbolic plane (here shown in the Poincaré disk projection). An ideal rotation is an isometry (or here a one-parameter group of isometries), which preserves only one ideal point: it is a limit between rotations and translations; its trajectories are called "horocycles", so here we see various points moving along horocycles.

The tiling is the set of fundamental triangles of the (2,4,5) tiling (i.e., each triangle has angles of π/2, π/4 and π/5 at its vertices).

Some triangles have been colored in such a way that the fundamental group of the pattern (i.e., the quotient of the group of isometries of the tiling by that of the colored triangles) is the symmetric group on 5 elements (so in particular, one triangle out of 2×5!=240 is colored in each color), and an order 3 isometry relates the three colors.

Hyperbolic kaleidoscope

David Madore 2013-12-04 | A uniform tiling of the hyperbolic plane (by regular quadrilaterals with 72° at each angle), colored periodically, being translated in various ways.

Visualizing the Earth in different projections

David Madore 2013-11-25 | Viewing the Earth's surface under five different azimuthal projections of the sphere:

1. orthographic (at 10s),

2. gnomonic (at 1m02s),

3. stereographic (at 1m54s),

4. Lambert azimuthal equal-area (at 2m46s),

5. azimuthal equidistant (at 3m38s).

[This video has no audio.]

For more comments and explanations, and also analogous projections of the hyperbolic plane, see: http://www.youtube.com/watch?v=xHvAqDuWG2M

Visualizing the sphere and the hyperbolic plane: five projections of each

David Madore 2013-11-19 | Visualizing the sphere and then the hyperbolic plane under various azimuthal projections (five of each).

spherical projections:

1. orthographic projection of the sphere (at 17s),

2. gnomonic projection of the sphere (at 1m16s),

3. stereographic projection of the sphere (at 2m33s),

4. Lambert azimuthal equal-area projection of the sphere (at 3m59s),

5. azimuthal equidistant projection of the sphere (at 5m23s);

hyperbolic projections:

6. Poincaré disk model of the hyperbolic plane (at 6m27s), analogous to stereographic,

7. Beltrami--Klein model of the hyperbolic plane (at 8m54s), analogous to gnomonic,

8. pseudorthographic projection of the hyperbolic plane (at 10m19s), analogous to orthographic,

9. Lambert azimuthal equal-area projection of the hyperbolic plane (at 11m20s),

10. azimuthal equidistant projection of the hyperbolic plane (at 11m55s).

The spherical projections show the sphere with a dodecahedral tiling by regular pentagons; the hyperbolic projections show the hyperbolic plane with a uniform tiling by regular heptagons.

The program used to compute the frames is available at http://www.madore.org/~david/programs/#prog_projections

For the same projections of the Earth's surface, see: http://www.youtube.com/watch?v=LKcTbIsWS9c

The sound of the Cantor set

David Madore 2013-10-17 | The sound signal whose Fourier transform equals the uniform distribution on the standard Cantor triadic set placed between 1056Hz (just/natural C6) and 3168Hz (just/natural G7). The graph shows the spectrum in question.

Rotation of the A3 lattice

David Madore 2012-08-13 | The lattice of the A_3 root system (A_3 lattice for short) goes by many names: it is also the D_3 lattice, the (vertex arrangement) of the tetrahedral-octahedral honeycomb, the face-centered cubic (or close-packed cubic) crystal system, or A-B-C dense sphere packing system. It consists of three alternating layers of plane hexagonal arrangements, or equivalently, two alternating layers of plane square arrangements; and it defines tetrahedral and octahedral cells. This video shows a set of thusly arranged spheres (323 of them, the closest neighbors of a central sphere) being rotated by one full turn.

Rotation of the E8 root system

David Madore 2012-02-13 | (See the description of http://www.youtube.com/watch?v=E-LC_l3gNuc for more about E_8 and its root system.)

The root system of E_8, an 8-dimensional polytope also known as the Gosset 4_21 polytope (here projected on a 2-plane that initially shows a 24-fold symmetry), is made to rotate uniformly, at a constant rate (technically: along a one-parameter subgroup). The polytope shown is always the same, it is merely rotated in 8-dimensional space, always in the same manner.

The rotation chosen here is special in that it belongs to the exceptional Lie group G_2 of automorphisms of the octonions. The E_8 root system forms the loop of units of (some copy of) the Cayley integral octonions (which are an E_8 lattice), and G_2 here refers to those rotations which preserve multiplication on the octonions (this restricts from the 28-dimensional group of all 8-dimensional rotations to a 14-dimensional subgroup). The one-parameter rotation subgroup was chosen randomly inside G_2 with the constraint that its angular velocities are in ratio of the golden ratio (so the video is never periodic, although it will come arbitrarily close to its starting position).

The vertices at the left and right which remain motionless throughout the video are the octonions −1 and 1, which are obviously fixed by G_2.

An interactive JavaScript version of this video is at http://www.madore.org/~david/math/e8rotate.html (requires a modern browser, but much prettier).

The beauty of E8

David Madore 2012-01-26 | The E_8 root system, or Gosset 4_21 polytope, is an exceptional uniform polytope in 8 dimensions, having 240 vertices and 6720 edges. This video shows a 2-dimensional projection of this polytope as it rotates in various ways.

The first 1′30″ of the video show various small rotations of the polytope to illustrate some of its highly symmetric plane projections (at 10″ we see a 30-fold symmetry known as the Petrie figure, at 20″ a 20-fold symmetry, at 30″ a 24-fold symmetry, at 50″ an 18-fold symmetry and at 1′10″ a 14-fold symmetry). The remaining 2′30″ of the video show a small sample of the 696729600 symmetries of the polytope in a different way: this time, we always return to an equivalent projection (every 10″), after some rotation which left the polytope symmetric.

Note that the polytope shown is always the same, it is merely rotated in 8-dimensional space.

It is unfortunate that compression causes the video quality to be so bad (especially in the second part of the video, where larger rotations are performed).

The E_8 polytope has 240 vertices, 6720 edges, 60480 triangular faces, 241920 tetrahedral three-cells, 483840 simplicial four-cells, 483840 simplicial five-cells, 207360 six-cells (of two different kinds, 69120 and 138240 of each, both being 6-simplices) and 19440 seven-cells (facets; 2160 being 7-simplices and 17280 being 4_11 polytopes). While it is not fully "regular" (there are only three regular solids in 8 dimensions, all boring), it is "uniform" and in many ways exceptional, being the largest of its kind. It is crystallographic in that its vertices span a lattice, the E_8 lattice, with many further remarkable properties (it is the only unimodular even lattice in dimension 8, the smallest nontrivial possible dimension).

More information at: http://en.wikipedia.org/wiki/4_21_polytope http://en.wikipedia.org/wiki/E8_lattice and http://www.madore.org/~david/math/e8w.html

Higher quality video download: http://www.madore.org/~david/misc/e8views.webm.torrent (or remove the .torrent extension if BitTorrent fails to work, but please try to use it if possible)

Transitive permutation groups on six objects

David Madore 2011-12-30 | There are, up to conjugacy, sixteen transitive permutation groups on six objects ("transitive" meaning they can move any given object to any given place). This video illustrates each of them, in turn, by selecting random elements from the group and making them act on six colored disks by moving them around. At one end, 𝔖_6 can rearrange the objects in any way whatsoever, and at the other end, C_6 can only permute them cyclically.

Visiting a Kerr black hole and entering the other interior region

David Madore 2011-03-31 | *** Watching http://www.youtube.com/watch?v=T_TU6T4-0LU before this video is recommended ***

Journey of an observer falling inside a(n ideal) Kerr black hole and emerging in a parallel universe. The black hole has a mass of roughly one million solar masses (Schwarzschild radius = 10 light seconds) and an angular momentum at 80% of maximality (a/M=0.8). The observer has an energy equal to its rest mass (i.e., just the escape energy) and angular momentum 1.3 (black hole mass times observer mass) along the black hole's axis: this angular momentum is sufficient to enter a different black hole interior (=region III) than if the angular momentum had been zero (as in the video http://www.youtube.com/watch?v=T_TU6T4-0LU with which compare).

The left portion is the observer's front view (for a somewhat arbitrary definition of "front"). The upper right box displays the trajectory on a polar plane cut (external horizon is red, internal horizon is green, static limit is dashed and is not seen in the video, cut discontinuity is purple, and trajectory is blue). The middle right box is a Penrose diagram (outer (I) blocks are shown in blue, inner (III) blocks are shown in pink, and intermediate (II) blocks are shown in light or dark grey according as they are white hole or black hole regions; the trajectory is again shown in blue). The lower right box shows the Boyer-Lindquist coordinates and their derivative with respect to the proper time (s) of the observer.

In the video, a blue sphere is placed outside the black hole at some distance, a purple sphere is placed in negative space (i.e., beyond the singularity cut), and the outer and inner horizons are various shades of red and green (red/orange/brown for outer, green for inner; lighter shades are white hole horizons, darker shades are black hole horizons). All spheres are checkered in an identical way, with twenty-four longitudinal stripes and twelve latitudinal (or polar) stripes, consistent with the black hole's axis. (The longitudinal stripes on the horizons rotate with the black hole.) The ring singularity itself is not visible as such, but appears as the edge rim of the purple region.

Note: The big black blob briefly seen when the observer is in the inner region (around 20″) is a computation artefact and not a real phenomenon.

More explanation, other videos and higher quality download ← http://www.madore.org/~david/math/kerr.html

Diving into a Kerr black hole

David Madore 2011-03-20 | *** Watching http://www.youtube.com/watch?v=T_TU6T4-0LU before this video is recommended ***

Journey of an observer falling inside a(n ideal) Kerr black hole and emerging in negative space. The black hole has a mass of roughly one million solar masses (Schwarzschild radius = 10 light seconds) and an angular momentum at 80% of maximality (a/M=0.8). The observer has an energy of 1.574 times its mass and angular momentum 0.243 (black hole mass times observer mass) along the black hole's axis: this energy is just barely sufficient to cross completely into negative space, hence the slowing down at about 25″ into the video.

The upper left portion is the observer's front view (for a somewhat arbitrary definition of "front"), the lower right portion is their rear view. The lower left square displays the trajectory on a polar plane cut (external horizon is red, internal horizon is green, static limit is dashed and is not seen in the video, cut discontinuity is purple, and trajectory is blue) and viewed from the top (the angle φ being that of Boyer-Lindquist coordinates). The upper right square shows the Boyer-Lindquist coordinates and their derivative with respect to the proper time (s) of the observer.

In the video, a blue sphere is placed outside the black hole at some distance, a purple sphere is placed in negative space (i.e., beyond the singularity cut), and the outer and inner horizons are various shades of red and green (red/orange/brown for outer, green for inner; lighter shades are white hole horizons, darker shades are black hole horizons). All spheres are checkered in an identical way, with twenty-four longitudinal stripes and twelve latitudinal (or polar) stripes, consistent with the black hole's axis. (The longitudinal stripes on the horizons rotate with the black hole.) The ring singularity itself is not visible as such, but appears as the edge rim of the purple region.

More explanation, other videos and higher quality download ← http://www.madore.org/~david/math/kerr.html

Orbiting a Kerr black hole (the last stable direct orbit)

David Madore 2011-03-18 | Journey of an observer orbiting a Kerr black hole at the last stable orbit in the direct direction (the direction in which the black hole is rotating). Here, the black hole has a mass of roughly one million solar masses (Schwarzschild radius = 10 light seconds) and an angular momentum at 80% of maximality (a/M=0.8).

The observer is on the equatorial plane, at a distance (r coordinate) of 1.453 Schwarzschild radii (4.4 million kilometers), has a total energy of 87.8% of their rest mass energy (the minimal possible value for a circular orbit around this black hole), and rotates around the black hole in 180.8s as seen by asymptotic observers, but 83.9s in their own proper time. The camera is stabilized by a gyroscope, but because of gyroscopic precession, it takes 155.7s (the length of this video) in the observer's time for it to return to the same point of view.

In the video, a blue sphere is placed outside the black hole at some distance (representing fixed distant stars), and the outer and inner horizons are various shades of red and green (red/orange/brown for outer, green for inner; lighter shades are white hole horizons, darker shades are black hole horizons). All spheres are checkered in an identical way, with twenty-four longitudinal stripes and twelve latitudinal (or polar) stripes, consistent with the black hole's axis. (The longitudinal stripes on the horizons rotate with the black hole.)

More explanation, other videos and higher quality download ← http://www.madore.org/~david/math/kerr.html