Greg Egan

@gregegansf.bsky.social

2265 Followers 114 Following

SF writer / computer programmer Latest novel: MORPHOTROPHIC Latest collection: SLEEP AND THE SOUL Web site: http://gregegan.net Also: @gregeganSF@mathstodon.xyz

Two electron micrographs of a collection of globular structures, with the second much sharper and higher in contrast.
The ring-shaped illuminating light (depicted in green in figure) that passes the condenser annulus is focused on the specimen by the condenser. Some of the illuminating light is scattered by the specimen (yellow). The remaining light is unaffected by the specimen and forms the background light (red). When observing an unstained biological specimen, the scattered light is weak and typically phase-shifted by −90° (due to both the typical thickness of specimens and the refractive index difference between biological tissue and the surrounding medium) relative to the background light. This leads to the foreground (blue vector in accompanying figure) and background (red vector) having nearly the same intensity, resulting in low image contrast.
On the left is a microscope image of some almost transparent cells, and on the right is an image of the same cells showing much more contrast and detail.
Seven pictures of the same tetrahedron, with different red/blue vertex colourings and arrows pointing from blue vertices to red ones, then a picture of the seven cones formed by those seven sets of arrows (4 sets of 3 arrows and 3 sets of 4 arrows).
Three pictures of the same triangle, with different red/blue vertex colourings and arrows pointing from blue vertices to red ones, then a picture of the three cones formed by those three pairs of arrows.
A set of 7 pyramids, four triangular and three square, that fit together to form a half-space.
layer of moss. The garden had been weeded, the azalea bushes neatly trimmed. No one had lived in the house for half a year, and a couple of days before they'd had a gardener come to whip the yard into shape. "Maybe this wasn't necessary, but around here, gardens are quite significant," Mr. Komatsu said. "Of course," I said, making the expected reply. "One other thing is that the persimmon tree bears wonderful fruit, but it's way too tart to eat, sad to say. But at least then the neighborhood children won't sneak into your garden to steal the fruit." "You're saying that everybody knows that? That the persimmon tree here might look nice but the fruit's too tart?" Mr. Komatsu nodded several times. "That's right. People around here know everything going on around here. Up to and including a single fruit on a permission tree." From “The City and its Uncertain Walls” by Haruki Murakami, translated by Philip Gabriel, Harvill Secker, London, 2024.
OP's figure, with added reflection in top side of solid triangle.
A table of knot diagrams, and the corresponding table of pictures based on a new knot invariant, which are hexagonally symmetric patterns of red and blue dots.
A red hyperboloid of one sheet, rotating. White curves spiral around it at a constant pitch, crossing cyan curves that wind around it ever faster.
A red hyperboloid of one sheet, rotating. White curves spiral around it at a constant pitch, crossing grey circles at ever increasing intervals away from the waist of the hyperbola.
A red hyperboloid of one sheet, rotating. Black curves run along the meridians of the hyperboloid, while grey curves form circles perpendicular to them.
A red hyperboloid of one sheet, rotating. Through one point on the waist of the hyperboloid, blue curves fan out around the hyperboloid then reconverge on the opposite side. There are cyan curves perpendicular to the geodesics and equally spaced along them.
A red hyperboloid of one sheet, rotating. Through one point on the waist of the hyperboloid, blue curves fan out around the hyperboloid then reconverge on the opposite side. Black curves fan out along the length of the hyperboloid, and two white lines at 45 degrees from the direction of the axis separate the blue and black curves.
The following text, over an image of an Old Testament era battle: prime HOUSE OF — DAVID Five feuding ballerinas are stranded at a deadly inn run by a reclusive ex-ballet prodigy and must... Watch now
An elliptical orbit, with vectors A and B joining one focus to the two points where the ellipse crosses the minor axis. The vector A+B is shown joining one focus to the other, while the vector A-B joins the two points on the minor axis.
Cover for the Korean collection “Sleep and the Soul” with a stylised image of a crescent moon made of dots and a figure falling through it.
A photograph of Einstein drawn as brightness contours.
A prolate spheroid (the surface created by rotating an ellipse around its long axis) with two neighbouring meridians running from the equator to the pole drawn in blue, and two red geodesics running on either side of the equator around 270 degrees before colliding.
A prolate spheroid (the surface created by rotating an ellipse around its long axis) with one meridian (an elliptical geodesic running from pole to pole) drawn in blue, the equator (a circle whose radius is the minor semi-axis of the ellipse) drawn in red, and a black dot with the word "Start" at a point where the meridian crosses the equator.
Heading: Tour website's AI sends visitors to Tasmanian sites that do not exist. AI-generated image of hiker walking along trail beside steaming body of water.
Estimated cancer deaths in Australia for 2025 In 2023, pancreatic cancer was the fourth most common cause of cancer death in Australia. It was projected to be the third most common in 2025. A graph with “Lung cancer”, “Pancreatic cancer”, “Liver cancer” and “Brain cancer” marked on the x-axis, with the y-axis ranging from 0 to about 9,000, and the graph itself comprised of line segments joining ten different points for specific cancers.
A red circular orbit around a small black sphere, and a blue circular orbit of the same size, tilted with respect to the first. These are at the bottom of the image, on the bottom face of a box, labelled "space", while the vertical edge of the box is labelled time. From points on each of the orbits, helices rise up in the time direction, wrapping around each orbit once as they climb to the top of the image.
A sphere, with one meridian coloured red and a nearby meridian coloured blue. The separation between them at the equator is labelled s(0), while the separation a distance d up from the equator is labelled s(d).
A collection of nested oval curves of increasing size, all tangent to the y-axis at the origin. The collection is mirror-symmetric about the y-axis.
A plot of e_xz versus e_yz, which is like one quadrant of a smoothed square. The curve is symmetrical in the diagonal e_xz = e_yz, and wherever one of the two eccentricities is greater than the critical value of 0.812, where the curve crosses the diagonal, the other eccentricity is less.
A triaxial ellipsoid is shown in cutaway to expose a series of concentric ellipsoids, the equipotential surfaces within. Initially these equipotential ellipsoids are not the same shape as the body they sit inside, but as the spin increases they expand and deform, until at an angular momentum of L = 0.38 they are all the same shape.
An oblate spheroid is shown in cutaway to expose a series of concentric spheroids, the equipotential surfaces within. Initially these equipotential spheroids are not as oblate as the body they sit inside, but as the spin increases they expand and deform, until at an angular momentum of L = 0.38 they are all the same shape.
A sphere coloured by longitude rotates ever faster, keeping the same volume as it deforms into an oblate spheroid (with two equal semi-axes), and then a tri-axial ellipsoid, with three different semi-axes. Initially a=b=c=1 for the semi-axes, I = 0.4 for the moment of inertia, U = -0.6 for the potential energy, with zero spin ω, angular momentum L, and kinetic energy K. At its fastest rotation, ω = 0.46854, I = 0.81103, L = 0.38, U = -0.55562, K = 0.08902, and the semi-axes are a = 1.83394, b = 0.83175, c = 0.65557
A fixed ellipsoid with a parallelepiped enclosing it, rotating and deforming so that its faces are always tangent to the ellipsoid at their centres.
A fixed ellipse with a parallelogram circumscribed around it, rotating and deforming so that its sides are always tangent to the ellipse at their midpoints.
There is an ellipse (blue) in a horizontal plane and a hyperbola (green) in a vertical plane, with the vertices of each curve coinciding with the foci of the other. A red-and-black striped string passes from a focus of the hyperbola to a point on the hyperbola, then an intermediate point, then a point on the ellipse, then a focus of the ellipse, with the string forming 4 straight line segments. The portion of the string that runs from the hyperbola to the intermediate point lies on a cone centred on the tangent to the hyperbola, and similarly the portion that runs from the ellipse to the intermediate point lies on a cone centred on the tangent to the ellipse.
There is an ellipse (blue) in a horizontal plane and a hyperbola (green) in a vertical plane, with the vertices of each curve coinciding with the foci of the other. A red-and-black striped string passes from a focus of the hyperbola to a point on the hyperbola, then an intermediate point, then a point on the ellipse, then a focus of the ellipse, with the string forming 4 straight line segments. For a sequence of choices of points on the ellipse, the point on the hyperbola is moved up from the plane, while the intermediate point traces out meridian-like curves on the ellipsoid, running from the plane to the point where the hyperbola intersects the ellipsoid.
There is an ellipse (blue) in a horizontal plane and a hyperbola (green) in a vertical plane, with the vertices of each curve coinciding with the foci of the other. A red-and-black striped string passes from a focus of the hyperbola to a point on the hyperbola, then an intermediate point, then a point on the ellipse, then a focus of the ellipse, with the string forming 4 straight line segments. For a sequence of choices of points on the hyperbola, the point on the ellipse is moved all the way around, while the intermediate point traces a loop on the surface of an ellipsoid. When the point on the hyperbola is in the plane of the ellipse the loop is a larger ellipse in the same plane. As the point on the hyperbola rises out of the plane, the loops that are traced grow smaller, until the last one is a small circle around the point where the hyperbola intersects the ellipsoid.
A close up of the head of a caterpillar, with four successively smaller versions of the same head forming a tower on top of it.
Banner: Kindle direct publishing Image: A dog chewing on a bar graph.
Amazon page for Ellipsoidal Figures of Equilibrium by Chandrasekhar (a book about the hydrodynamics of rotating stars) categorised under Books › Arts & Photography › Graphic Design › Commercial
Photo of front cover and spine of Russian translation of “The Clockwork Rocket”, showing a rocket engine firing against a backdrop of colour-trails of stars.
A change of variable on an integral from θ to u = sin θ, du = cos θ dθ. The original integral has a cos θ dθ in the numerator, which is correctly changed to du, but as well as substituting u for sin θ elsewhere in the integral, a factor of 1/√(1-u^2) is introduced, effectively dividing the integrand by cos θ for no reason.
Mathematica input: Assuming[R > 0 && M > 0 && 2 M < R, FullSimplify[ Series[ Sqrt[2] R ArcTan[(Sqrt[R] - Sqrt[R - 2 M])/Sqrt[2 M]] - Sqrt[M (R - 2 M)], {M, 0, 2}]]] Output: (4*M^(3/2))/(3*Sqrt[R]) + O(M)^(5/2)
Four panels, each showing a square with a black line, containing two black dots, cutting across a different one of its four corners. In each case, there is a red dot at the corner of each square horizontally opposite the cut-across corner, and along with the two black dots and one more red dot they form a grey parallelogram. A red line extending the edge of the parallelogram containing the second red dot and one of the black dots runs between the two opposite vertical sides of the square. There is a blue dot in the corner diagonally opposite the corner with the red dot, and a dashed blue line joins it to the black dot that lies on the square-crossing red line.
A small sphere moves around inside a larger one, with a tetrahedron fitting between the two.
Pairs of random points appear inside a dodecahedron, and a line joining them intersects pairs of faces, which turn red, green or blue depending on the angle between them.
The Reasoning The core of the argument lies in a specific, non-obvious geometric property of the regular 600-cell. 1. Opposite Cells: The regular 600-cell is a centrally symmetric 4-dimensional polytope. Its boundary is composed of 600 identical regular tetrahedral cells. For any given tetrahedral cell on the boundary, there is a corresponding "opposite" cell located directly across the center of the 600-cell. These two cells lie in parallel 3-dimensional hyperplanes. 2. The "Corridor" between Opposite Cells: Consider the convex hull of any pair of opposite tetrahedral cells. This forms a small 4-dimensional region— a "corridor" —that connects the two cells through the interior of the larger 600-cell. 3. The Crucial Geometric Fact: For the regular 600-cell, the polytope "curves" away from this corridor so sharply that the corridor lies entirely within the interior of the 600-cell, only touching the boundary at the two opposite cells themselves. Consequently, any straight line that intersects the interior of one of these opposite cells cannot also intersect the interior of the other. A line can only touch both cells if it passes exclusively through their boundaries (e.g., through an edge of one and a vertex of another). 4. Probability of Hitting the Boundary: When two points are chosen uniformly at random from the interior of the 600-cell, the line they define is also random. The probability of such a random line passing precisely through lower-dimensional elements like vertices, edges, or the faces of the boundary tetrahedra is zero. For the line to "intersect" a cell in a way that has a non-zero probability, it must pass through its 3-dimensional interior. Since a single straight line cannot pass through the interiors of two opposite tetrahedral cells of a regular 600-cell, the probability of this event occurring is zero.
Question: If I pick two points uniformly at random in the interior of a regular 600-cell, what is the probability that the line containing those two points will intersect any two tetrahedral cells on the boundary of the 600-cell that lie in parallel hyperplanes? Answer from Gemini 2.5 Pro: The probability is zero. This is a fascinating problem in geometric probability concerning the structure of a 4-dimensional object. The result is highly counter-intuitive, as one might expect a line passing near the center of the 600-cell to connect two opposite faces. However, the specific geometry of the 600-cell leads to this surprising conclusion.
Admiralty Gnomonic Chart of the Indian and Southern Oceans, for use in plotting great circle tracks
A large rodent with damp brown fur is seen halfway through entering a body of water from the bank.
A cube sliced in two, with the surface along the cut forming a parallelogram.
Two vectors of equal length are shown, labelled A and B, pointing from a common starting point, labelled "First focus", which is the right-hand focus of an ellipse with a horizontal major axis. The endpoints of the vectors A and B lie at the top and bottom points of the minor axis of the ellipse. The vector A+B is shown, pointing from the right-hand focus of the ellipse to the left-hand focus, and the vector A-B is shown, pointing from the endpoint of A at the top of the ellipse down to the endpoint of B.
Truncated icosahedron soccer ball (32 faces, 20 white hexagons, 12 black pentagons)
Animation starts with a 9 by 9 square of right-triangles of 6 dots, each with rows of length 3, 2 and 1 from bottom to top, right-justified. The triangles on or below the square’s diagonal have all black dots, while the triangles above the square’s diagonal have 3 cyan dots below their hypotenuse, and 3 yellow or red dots along their hypotenuse, with yellow dots for the first four slanted rows closest to the square’s top left corner, and red for the rest. To the bottom left of the 9 by 9 square is a right-triangle of 10 green dots, with rows of length 4, 3, 2 and 1 from bottom to top, also right-justified. As the animation proceeds, all the green dots and black dots stay fixed. The cyan dots are reflected across the square’s diagonal to fill in the gaps beside the black dots to make the black triangles into complete squares, while the yellow and red dots move down to fill in the space to the right of the green triangle, so that the final image is a complete right-triangle.