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Matt Henderson

@matthen282k followers

maths, visualisations, conversational AI. VP Research @polyaivoice prev: @RekaAILabs, @Apple AI/ML, @GoogleAI, PhD @Cambridge_Eng

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1Apr 1, '23to tell if a maze is solvable, just hang it by its corners! The first maze stays in one piece, so there is no path from the entrance at the top to the exit at the bottom. The secon78×3.4M23k1.5k1.6k
2May 4, '24a thread with some of my top animations 📍🧵37.1×1.6M6.6k4802.8k
3Jul 2, '22give each pixel a random Pokemon type, and then battle pixels against their neighbors, updating each pixel with the winning type (using the Pokemon type chart) we quickly see areas31.5×043k5.8k4.1k
4Oct 13, '23one easy way to make a simulation of falling sand piling up- just define what happens for all 16 possible 2x2 grids, and repeatedly apply these rules to the picture24.2×1M6k5151.4k
5May 30, '20we can model a lightning strike by finding the shortest path in a random maze, from a point at the top to the ground. To find the path, we send out a frontier through the maze, and22.9×030k6.8k1.3k
6Sep 21, '21the dumbest way to solve a maze? simulate a gas of thousands of particles diffusing from the start point, until one particle reaches the exit. trace back the winning particle19.8×030k3.8k871
7May 3, '20a method of drawing infinitely many touching circles15.3×020k4.2k879
8Oct 11, '23an exponential curve looks identical anywhere along the x axis- so long as you stretch the y axis accordingly13.1×570k2.6k212255
9May 26, '20when an object is pulled out of this magical lake, its reflection comes with it. We can make a 3D fractal through repeatedly rotating, submerging, and scaling an initial cube11.4×016k3k498
10Apr 3, '24a simple way to fingerprint a language- take a bunch of text in the language, and every time one character occurs after another, connect them with a line from the left side to the 10.6×460k3.2k2581.2k
11Jul 31, '21visualizing prime factors10.1×013k2.7k890
12Nov 27, '22watch a 2 layer neural network learn to separate two classes to the left and right9.8×013k1.6k1.6k
13Apr 30, '23balls bouncing in a circle appear to be curving. But they are actually going in straight lines (keep watching)9.4×407k3.6k528300
14Apr 12, '23can a neural network learn to walk as a physical object in a physics simulation? here I train walking neural nets with an evolutionary algorithm. The input nodes/feet are activated7.8×339k1.8k236286
15Sep 4, '22de-impressionizing a Monet7.5×08.2k785608
16Feb 16, '24can you see the spinning dodecahedron? pause the video and it disappears7.0×304k2.2k345453
17Apr 11, '20a parabola is formed when circles radiating from a point meet lines moving at the same speed. this is equivalent to slicing a cone parallel to its slope6.7×09k1.7k433
18May 15, '21isn't it surprising that such a simple system is chaotic? Two balls start at almost exactly the same point, but their trajectories rapidly diverge after only a few bounces6.6×09.4k1.3k428
19Feb 20, '22I was a bit surprised that you can post a Pringle through a letterbox- because twisting straight lines trace out the hyperbolic paraboloid Pringle shape6.1×08.8k1.2k379
20Jun 5, '20a simple and random process that creates an intricate and structured fractal pattern. At each step, pick a random edge to form a triangle, and the next point is the center of the t5.9×07.4k1.7k407
21Apr 17, '21zoom in5.6×08.4k1.1k228
22Apr 9, '23Minesweeper, but on the new Einstein tiles5.6×242k2.8k306137
23Sep 4, '21a visualization of chaos- bouncing on a sine wave the predictions for the ball's trajectory (orange) quickly diverge after a few bounces5.4×07.8k1.1k459
24Apr 13, '20draw a line, and extend it deflected by a fixed angle. Continue, adding the angle again each time. The result is a Euler spiral4.9×06.5k1.3k316
25Mar 17, '24hello world in python... using a genetic algorithm4.5×196k2.5k185689
26Apr 30, '22a nice way to see how the learning rate affects Stochastic Gradient Descent. we can use SGD to control a robot arm - minimizing the distance to the target as a function of the angl4.4×05.3k871909
27May 7, '20approximate patterns traced by the inner planets, relative to the Earth4.4×05.8k1.2k199
28Oct 27, '23a fun way to draw a picture from the digits of pi, made from circular arcs. draw an arc that goes 3/10 of the way around the circle. Now flip the circle over and continue the curve4.3×187k1.6k221267
29Mar 26, '23prime factorization trees4.3×187k2.2k327250
30Apr 30, '21'the butterfly effect' this water wheel is a simple system that demonstrates chaos theory. As it spins seemingly randomly left and right, its center of mass (orange) traces out a b4.2×05.4k1.1k431
31Mar 26, '23playing with ways to visualize the factorization of numbers4.1×177k1.6k216202
32Sep 14, '218 Earth years are roughly equal to 13 Venus years, meaning the two planets approximately trace out this pattern with 5-fold symmetry as they orbit the Sun4.0×04.8k1.1k317
33Jul 9, '20a pringle is a hyperbolic paraboloid4.0×05.2k1.1k144
34Apr 10, '21Cruithne has been nicknamed 'Earth's second moon'. Its elliptical orbit around the Sun is in resonance with the Earth's, so that from our perspective it appears to be following a b3.9×05.6k764267
35Jun 25, '20a slice through a cylinder creates an ellipse. when unrolled, it creates a sine wave3.6×05.1k810203
36Dec 15, '21constructing a cube from a hexagonal ring of squares3.6×05.3k668285
37Apr 14, '21this gas, modelled with boucing balls, spreads out equally between the two containers. This would normally be irreversible, but I have chosen the initial velocities carefully...3.5×05k653246
38May 13, '20why the derivative of sin is cos3.5×04.6k898265
39Dec 17, '22each dot (except the one in the middle) eventually escapes to infinity... yet the animation loops perfectly?3.5×151k1.5k129141
40Dec 12, '21two balls rolling down two slopes. while the indirect path proves faster at first, the direct path wins in the end3.3×04.7k437398
41Sep 25, '21every triangle is an equilateral triangle (when viewed from the right perspective)3.2×05k471161
42Oct 31, '21an inverse square law for gravity gives elliptical orbits. but what about an inverse r^2.01 law?3.0×04.4k544216
43Apr 21, '20an explanation of refraction. When a wave changes its speed and wavelength at a boundary, it must change its direction so that the amplitudes match up at the boundary2.9×03.6k804234
44Jun 12, '22making a fractal at home, with two mirrors and two phones2.9×04.3k508211
45Aug 15, '24plotting the four energy terms of a double pendulum as a stacked bar chart. the total height of the bar is constant (conservation of energy), though each individual part behaves ch2.9×124k1.8k192378
46Nov 21, '21bouncing from the focus of one parabola to another2.7×04.3k416159
47May 22, '20if a black hole passed in front of a mirror, what would its reflection look like?2.7×03.9k56590
48May 2, '21a large image of hundreds of flocking birds contains enough information to create an animation of a single bird flying2.6×04.2k366138
49Sep 15, '21did you know that a parabola, when viewed from the right perspective, appears as a circle?2.6×04.1k333165
50Jul 15, '20creating an infinite spiral zoom using complex analysis. The map z -> z^(x + i y) warps the image so that it can be infinitely repeated inside itself2.6×03.5k606212
51Jun 24, '20hyperbola, parabola, ellipse, circle- all are slices of a cone2.4×03.3k605183
52Oct 30, '21no matter how chaotic and unstable the orbits of these three gravitational bodies are, their centre of gravity (centroid of the triangle) always remains stationary and fixed in spa2.3×03.2k438238
53Aug 8, '21a fourier series- but using triangular waves and squares, instead of sine waves and circles2.3×03.2k480194
54Nov 14, '20blur and sharpen an image, again and again. After a while, the fabricated details form Turing patterns2.3×03.3k400197
55Jun 8, '20a visual proof that the volume and surface area of a torus is the same as a cylinder (with radius r and height 2πR). Each swivel does not change the volume or surface area2.1×03.1k47684
56May 18, '20a ring built from the Julia sets that lie along the main cardioid of the Mandelbrot set2.1×02.8k487137
57Aug 6, '21did you guess what shape is hidden in the circumference of this wire?2.0×03k319157
58Jan 17, '22imagine we have a stream of pixels coming in one by one. They form a video, but we don't know the width and height of each frame. Here we try a bunch of guesses for the dimensions,2.0×02.9k250232
59Jun 1, '21in a perfectly rectangular pool table, balls bounce in an orderly pattern. But even a slight curve in the walls will make it chaotic, so that after a few bounces the balls totally 2.0×03.1k332102
60Apr 14, '20construction of a Julia set through repeated twists, reflections, and shifts. The twists correspond to the complex square root.2.0×02.5k561125
61May 16, '20find the centroid of a triangle, by connecting the corners to the midpoints. this construction creates 6 new triangles. repeat1.9×02.7k457131
62Apr 8, '20an explanation of the arms in spiral galaxies, proposed by Lin and Shu. The arms are just denser regions, arising from slightly elliptical orbits with varying orientations1.9×02.2k615133
63Nov 17, '21a method to draw infinitely many touching circles -- I discuss this and a few other geometrical animations in the latest @Numberphile video:1.9×02.6k368167
64Feb 27, '21Using one cellular automaton to feed another code:1.8×02.6k358168
65Sep 19, '22interesting patterns emerge from balls bouncing inside rectangles at 45°. It's immediately obvious when the sides are rational vs irrational1.8×02.6k285211
66Jul 18, '20sorting the pixels within a photograph results in some cool effects. Here each vertical column of pixels is sorted by its hue1.7×02.7k29988
67Apr 11, '21balls bouncing inside a circle- they slowly disperse, creating cusps and spirals over time1.7×02.7k23273
68Jun 13, '20revealing the cube's hidden hexagon1.6×02.4k36477
69May 23, '20images as bar charts of red, green, and blue1.6×02.5k29279
70May 17, '20surprising patterns from slicing the Menger sponge fractal1.6×02.2k40988
71Mar 1, '22a julia set fractal can be created with repeated twistings and translations of a starting image. The twistings double the detail in the picture, and correspond to complex square-ro1.5×02.1k322138
72Jun 19, '20the modes of oscillation for a sphere- the spherical harmonics. These are closely related to the shapes of atomic orbitals1.4×01.9k363103
73Apr 17, '20modes of oscillation for a sphere- the spherical harmonics1.4×01.7k375128
74Mar 30, '21Dirac's belt trick is a clever way to remove two full twists in a ribbon by simply moving it around. This gives a way of connecting cables to a rotating object without them twistin1.4×02k278130
75Nov 30, '21finding the radius of curvature of a curve at a point, by finding the intersection of two normals very close to the point1.4×02k229138
76Apr 3, '21a pendulum, double pendulum, 4-, 8-, and 16-pendulum- all with the same total mass. I originally thought in the limit it would behave like a hanging rope — why doesn't it?1.3×01.7k248147
77Dec 3, '21take a vertical row of pixels from a video, and stack them to make a still image- here we can measure the frequency of the blinking light, and also pick up scans of the cars that d1.3×01.9k172164
78Sep 2, '21it's a good idea to make sure that the number of teeth of touching gears don't share a common factor. Here the defect on the 7-toothed gear is spread evenly over the 19-toothed gea1.2×01.9k198103
79Apr 3, '21a visual demonstration of the connection between the triangular numbers, and counting all possible pairs in a set1.2×01.7k265105
80Aug 24, '21new video on @numberphile discussing 'The Lightning Algorithm'1.2×01.6k220167
81May 23, '21as a planet orbits, its velocity traces out a circle around it. This fact was key in Newton's geometric proof of how bodies orbit under gravity1.2×01.7k20694
82Nov 27, '22check out what I found on google earth1.2×01.7k17799
83Apr 15, '20draw a random polygon, and connect the midpoints of its edges to make a new polygon. Repeat this and it converges to an ellipse1.2×01.5k31467
84Dec 24, '2010 years ago I started my maths animation blog on tumblr1.2×01.8k17093
85Apr 27, '20composing circular motion of different frequencies (columns) with harmonic motion of different frequencies (rows). All patterns are centered on zero except the ones on the diagonal1.2×01.4k337107
86Jun 10, '20the classic Mandelbrot set fractal is generated by the iteration z → z² + c. By increasing the power of z, we can create the 'Multibrot' fractals, with increasing rotational symmet1.1×01.5k31065
87Aug 29, '22could we use neural networks to make games look photo realistic? here I apply stable diffusion to minecraft, with the guiding text "beautiful photograph of outdoors". Pretty trippy1.1×01.5k183146
88Apr 5, '20geometric interpretation of circle inversion1.1×01.4k29288
91May 20, '20landscapes of arbitrary detail and expanse can be generated by layering random noise of increasing frequency and decreasing amplitude1.0×01.5k18589
92Sep 3, '22an app where kids can view their lego models as if they were real and full scale I made these with Stable Diffusion - pushing the photo away from 'lego' and towards 'photograph'1.0×01.4k150113
93Sep 20, '21just as a 3d cube can be unfolded to a net of 2d square faces, a 4d tesseract can be unfolded to a net of 3d touching cubes1.0×01.4k21569
94Nov 28, '20Creating an ellipse from a pile of sugar. The pile forms a rough cone, and then the hole underneath creates an upside down cone of missing sugar. The two cones intersect at an elli0.9×01.4k16950
95May 7, '20a hexagonal tile turns blue at each step if it has exactly one blue neighbour0.9×01.4k17738
96May 9, '21creating a heart-shaped cardioid from circular motions0.9×01.4k15447

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