Pictures that don't just show it's true — they show why. Anything with a halo can be dragged.
The dot F is the focus. The line is the directrix. The purple circle passes through F and just touches the line. Pump it up and watch where its centre goes.
Both red segments are radii of the same circle, so they are equal. The one to the line is the shortest way down, because a radius meets a touching line at a right angle.
XF = XD
A ray falls straight down onto the curve at X. Bouncing like a mirror means it leaves at the same angle it came in: angle ① = angle ②. If that's true, it leaves heading straight for F.
Why ① = ②: fold the picture along the green tangent. X is exactly as far from F as from D, so the fold lands D right on F. Without the mirror, the ray would go straight on to D. The fold flips that straight-on path onto XF, and folding doesn't change angles.
Click anywhere. A dot moves straight toward the parabola, parallel to its axis, and bounces. Every drop from inside the cup goes through F, then bounces once more and leaves straight up.
A pencil lies on side AB, point toward B. It turns at A through angle A, slides to C, turns through angle C, slides to B, turns through angle B, and slides home.
It ends on the same line but pointing the opposite way. Sliding never turns it, so it turned a total of A + B + C, and the dial shows the three pieces making a straight line.
Spin the triangle a half‑turn around the middle of side BC. The copy fits against it perfectly and makes a parallelogram. Slide parallelograms around and they tile everything.
Look at corner A. Above the straight line through A and B sit three corners, one of each colour: A, then C, then B. Together they fill a straight line.
A + C + B = 180°
Spin the shape a half-turn around the middle of one of its sides, then keep going. The copies fill the whole floor with no gaps, even for a lopsided shape, and even one with a dent. Drag the corners and see!
Look at corner A: four copies meet there, one of each colour. So the four angles make one full turn.
A + B + C + D = 360°
Pick one corner and draw lines to all the others. A shape with n corners splits into n − 2 triangles, and their corners exactly fill the shape's corners.
Just like the triangle: the pencil lies on the first side. At each corner it turns through the inside angle, then slides to the next corner. When it gets home it lies on the first side again, so it has turned a whole number of half-turns. The dial counts them: n − 2 half-turns, which is (n − 2) × 180°.
A walker goes along the edges. At each corner they turn by the outside angle, just enough to face the next edge. When they get home they face the same way they started, having turned around exactly once: 360°.
Each inside angle is 180° minus its outside turn. So the inside angles add up to n × 180° − 360° = (n − 2) × 180°, the same answer as the triangles!
(With a dent, the walker turns backwards at the dented corner, shown in grey on the dial, and the total is still exactly one turn.)
The source says “hee” 4 times a second, and each hee starts a ring that spreads at the speed of sound. You hear a hee each time a ring washes over the mic. When the rings arrive faster, the hee plays faster, so it's higher and shorter, exactly like speeding up a recording. (Real sound has 440 rings a second at this pitch, far too close together to draw.)
When the source moves, it chases the rings in front of it and runs away from the ones behind. In front they get squashed together, so more rings reach you each second: higher pitch. Behind, they're stretched out: lower pitch.
At v = c the gap (c − v)T is zero. Every ring piles onto the same wall in front of the source: that wall is the sound barrier.
Go faster and the source outruns its own sound. The ring edges form a cone, and you hear nothing until it sweeps over you. BOOM. After that you hear two sounds at once: the one it's making now, sliding down, and the one it made while it was coming toward you. That one arrives in reverse order, so it plays backwards.
Shape: click to add corners, click the first corner (or double‑click) to finish.
The tile is made from two arrows out of one corner, u and w. Slide the tile by u, or by w, as many times as you like, forward or backward. That's the whole pattern.
Escher's trick: draw a shape that sticks out past one edge. The neighbouring copy makes it poke in through the opposite edge, so a bump on one side is a dent on the other.
The dashed square is the whole picture. Each coloured box is a smaller copy of the whole picture, moved, turned or squashed however you like. The little F shows which way is up.
One round of the machine: shrink a copy of the picture into every box, then throw the old picture away. Do it again and again (turn up Depth). The fractal is the picture that stops changing, because it is made exactly of shrunken copies of itself.
Animate rounds plays the machine slowly, with any boxes you make. There are two ways to watch a round. Refine each piece: every piece splits into smaller pieces arranged just like the boxes. For Sierpinski that's "cut the middle out of every triangle". Copy into boxes: the whole picture shrinks into each box.
Dots: start anywhere and jump into a random box, again and again, leaving a dot each time. The dots fill in the same fractal. The colour of each dot shows which box it just jumped into.
Stick two pins in the paper, F₁ and F₂. Tie a string between them, pull it tight with a pencil, and go all the way around. The curve you draw is an ellipse, and the two pins are its foci.
Light from F₁ hits the ellipse at P and bounces. The green line is the tangent. Angle ① (F₁P to the tangent) equals angle ② (PF₂ to the tangent), so the bounce goes straight to F₂.
Why: slide Q along the tangent. Every Q other than P is outside the ellipse, so F₁Q + QF₂ is longer than the string. That makes P the shortest way to go from F₁ to the line and on to F₂.
The Sun pulls the planet toward itself, harder when it's close: at half the distance the pull is 4 times as strong. Drag the planet, or the tip of its arrow to change how fast it's going, then let go.
The planet's path follows the dashed ellipse, and the Sun sits exactly on one of its pins! The string check still works: planet-to-Sun plus planet-to-other-pin never changes.
Tap anywhere to send a hee out of F₁ in that direction. It bounces to F₂, then back through F₁, again and again.
The area of a shape is how many little 1‑by‑1 squares it takes to cover it exactly.
A slanted rectangle is called a parallelogram. Cut off the triangle on one end and slide it to the other end. It fits perfectly and makes a rectangle.
Make a copy of the triangle and spin it a half-turn around the middle of one side. The two triangles make a parallelogram with the same base and height.
Sides don't have to be whole numbers. Cut every square into 10 strips, and every strip into 10 little squares. Each little square is one hundredth of a square: 100 of them fill one whole square.
No squares fit a curvy shape exactly. So fill it with thin rectangles that fit under the curve. They miss a little bit at the top. Make them thinner, and they miss less and less.
The knife only went part of the way. Can you always finish the cut with one straight line so both pieces are exactly the same size?
To measure a curvy piece, use the trick from Curvy: fill it with thin rectangles, then make them thinner and thinner.
Cut a pie into equal slices. The bottom number says how many slices there are. The top number says how many are shaded. So 3/4 means 3 of 4 equal slices.
This pie is cut into 100 thin slices. The bold lines cut it into 10 tenths. 0.25 means 2 tenths and 5 hundredths. That's 25 hundredths: .ab really means "ab hundredths".
The fraction wall is the pie unrolled into strips, cut into 1, 2, 3, … 10 equal pieces. The orange line is your decimal: drag it! Each row says how many whole pieces fit before the line. If the line lands right on a crack, that row gets an orange box: its fraction is exactly your decimal.
3/4 or 5/6? Slices of different sizes are hard to compare. So cut both pies into slices of the same size, then just count.
Lovey (the sloth) is very slow, so he wants the shortest way to the ice cream truck. Tap spots one after another to plan his route, starting next to Lovey. The numbers are how long each path is.
Zigzag rocky trails are slow, so they're longer than they look. Curvy vines are quick swings, so they're shorter than they look. Sloths love vines!
Every path is a little stream bed. Pour water at Lovey's spot, and it flows along every path at the same speed: 1 unit of path per tick. When water reaches a spot for the first time, it spills into all of that spot's paths.
The time water first reaches a spot = its shortest distance from Lovey.
Dijkstra's algorithm is the water without the waiting. Each dry spot next to a wet spot gets a note: "water could get here by ___" (a wet neighbour's time plus the path). The spot with the smallest note is the next one to get wet, and its note is exactly right, so we freeze it. Then update the notes and repeat.
Take a cone (actually two, point to point) and slice it with a flat knife. The edge of the cut is a curve, and which curve depends only on how much you tilt the knife compared to the cone's slope.
Flat: a circle. Tilted a bit: an ellipse. Tilted exactly as steep as the cone's side: a parabola (it never closes up). Steeper still: a hyperbola, which cuts both cones and has two pieces.
Drop a ball into the cone above the knife and another below it, each just big enough to touch the cone all the way around and touch the knife. They touch the knife at exactly the two pins of the ellipse, F₁ and F₂!
Why: from a point P on the ellipse, go straight along the cone toward the tip. You hit the small ball's touching circle, then the big one's. Two paths that start at the same point and touch a ball at their other end are the same length. So PF₁ = the red piece and PF₂ = the blue piece. Together they're the stretch of cone between the two circles, which is the same for every P. So PF₁ + PF₂ never changes: it's an ellipse!
Count the corners (V, for vertices), the edges (E) and the flat faces (F). Tap corners and faces to mark the ones you've counted. Then work out V − E + F.
V − E + F = 2
It's 2 for the cube, 2 for the bucky ball, 2 for every one of them. Euler noticed this in 1750. But why?
Imagine the solid is made of stretchy rubber, sitting on the table. Cut out the bottom face (orange) and stretch that hole wider and wider until the whole thing lies flat on the table. The hole becomes the outside. Tap a different face to cut that one out instead.
Start with a single dot. That's 1 corner, 0 edges, and 1 region (all the paper around it): 1 − 0 + 1 = 2. Now add the edges back one at a time. There are only two kinds of move:
A line to a new dot: +1 corner and +1 edge. These cancel, so V − E + F stays the same.
A line between two dots already there: +1 edge, and it cuts a region in two, so +1 face. These cancel too!
Or go backwards, like in the video: rub out edges one at a time. Erasing a line between two regions joins them (E −1, F −1). Erasing a dangling line takes its dot with it (V −1, E −1). You always end with one dot: 1 − 0 + 1 = 2.
Tilt the flat drawing so no edge is exactly flat, with one highest dot and one lowest dot. Now give out charges: +1 to every dot, −1 to every edge, and +1 to every face, including the big outside face. Adding up all the charges gives exactly V − E + F.
Now slide every charge (except the highest and lowest dots) into the face just to its right, one face at a time, and watch what happens.