Engineers call this shape a 'flying wing'. People dreamed of it since the 1920s: Jack Northrop, whose company built the B-2, flew flying-wing bombers in the 1940s. They were hard to keep steady, and the project was dropped. Fifty years later, computers solved that problem (more on that next).

Only 21 B-2s were ever built, and each cost about $1.16 billion in 1998 dollars, more than its weight in gold. They are based at Whiteman Air Force Base in Missouri and are kept in climate-controlled hangars, because the special skin doesn't like rain, heat or dust.

Drag is air resistance: the push of air against a moving object. A normal plane's body and tail make drag but almost no lift. Removing them is one reason the B-2 can fly about 11,000 km without refuelling.

The middle of the wing is deep enough to hold four engines, big fuel tanks and two bomb bays that can carry over 18 tonnes.

The downside? A flying wing has nothing at the back to keep it pointing straight, like an arrow without feathers. On its own it would wobble and swing. Keep reading to see how it stays steady.

Radar works by catching echoes, a bit like a bat. Big flat sides that face the radar, round tubes, and corners where two surfaces meet all send strong echoes back. A normal airliner has all three.

The B-2's designers started with the question 'what shape sends back the least?' and built the plane around the answer. Flying well came second, which is why it needs computers to fly at all.

These are called drag rudders. Opening both at once slows the plane down, so they also work as air brakes for landing. In flight the computers usually keep them slightly open on both sides, ready to open one a bit more or less.

Why not just add a small fin? Even a small fin is a flat, upright surface, and an upright surface meeting a flat wing makes a corner, one of the strongest radar echoes there is. The split rudders tuck flat when closed.

When a flap tilts down, it bends the air below it downward more, so that part of the wing gets more lift and rises. Tilting it up does the opposite. Using different amounts on each side rolls or pitches the plane.

The beaver tail (officially the gust load alleviation surface) moves to soak up gusts of wind, so a sudden bump doesn't bend the huge wing too much. It also helps trim the nose up or down.

There are three elevons on each side. Because there's no tail, each one has to do several jobs at once, and only a computer can work out the right mix in time.

This is called fly-by-wire: wires and computers replace the old cables and rods between the stick and the flaps. The computers measure the plane's speed, angle and turning many times a second and adjust before a wobble can grow.

There are four computers so that if one fails, the others carry on. But they all depend on good data. In 2008 a B-2 crashed on take-off in Guam because moisture got into the sensors that measure air pressure. The computers were fed wrong numbers, pitched the nose up too far, and the plane stalled. Both pilots ejected safely.

Radio waves are the same kind of energy as light, just with much longer waves: from a few centimetres to many metres. They pass through clouds, fog and darkness, which is why radar works at night and in storms.

An airport radar dish sends around a thousand bursts each second and spins once every few seconds. Each burst lasts about a millionth of a second, then the dish spends the rest of the time listening.

Why 150 m and not 300 m? Light travels 300 m in a millionth of a second, but the echo has to go out and come back, so the plane is half that distance away.

How strong the echo is depends on the plane's 'radar cross-section' (RCS): how big it looks to radar, not to your eyes. A shiny, round airliner might look like a flat metal plate tens of square metres in size. A stealth plane is designed to look far, far smaller.

Most airliners also carry a transponder, a radio that answers the radar with the flight's ID and height. A bomber trying to hide simply leaves it switched off.

Every radar receiver hears a constant hiss of random noise, from heat in its own electronics and radio waves from space. An echo only counts if it stands out above that hiss.

Here's the surprising maths. A radar wave spreads out on the way to the plane, and the echo spreads out again on the way back. So the echo gets weaker with distance × distance × distance × distance. That means making the echo 10,000 times weaker only shrinks the detection range 10 times.

That's why stealth has to be extreme. Cutting the echo by 10 times only makes the radar's range about 1.8 times shorter. Designers aim for many thousands of times less echo than a normal plane of the same size.

Only a surface that faces the radar head-on sends a strong echo straight back. Tip it a few degrees and the echo leaves at an angle and misses the dish completely.

Edges are tricky too: a sharp edge scatters a little radar in many directions. That's why the B-2's front edges are swept back at 33°, so the edge only 'faces' radars off at an angle, never one dead ahead.

This is called planform alignment. 'Planform' is the shape seen from above. Because the edges line up, their small echoes all pile into the same few directions instead of being spread around the compass.

The four beams point at fixed angles, about 33° either side of the nose and of the tail. A radar dead ahead of the B-2, the one most likely to be waiting for it, sits between beams, in the quiet zone.

Even the edges of panels and doors (the bomb bay and landing gear doors) are zig-zagged at the same angles, so they don't add echoes in new directions.

Engineers call this a corner reflector. Two walls at right angles send the wave back along its own path, whatever angle it arrived at. Sailors put metal corner reflectors on small boats on purpose, so ships' radar can see them.

That's why the B-2 has no tail fin, why its body is blended into the wing with smooth curves, and why the cockpit windows have a thin metal coating: otherwise radar would get inside and echo off the seats and instruments.

The engines are General Electric F118s. Each pushes with about 77,000 newtons (17,300 lb) of thrust, roughly the weight of five family cars. Together they push a plane that can weigh over 150 tonnes to just below the speed of sound.

They have no afterburner, the extra flame some fighters use for a burst of speed. An afterburner would make a huge, bright heat signature that infrared sensors could spot from far away.

What if an engine fails? The B-2 can keep flying on fewer engines, and because they are side by side in pairs, the push stays fairly balanced.

The fan face is a disc of metal blades, a perfect echo maker. Engineers also worry about the blades spinning: they add a flicker to the echo that can even reveal what type of engine it is.

The intakes sit on top of the wing, so radars on the ground below can't look into them at all.

If each bounce soaks up 90% of the wave, then after three bounces only 0.1% is left. The S-bend forces several bounces before a wave could reach the fan and come back out.

The air flowing to the engine must stay smooth, though. A badly curved tunnel would make swirls that could stall the engine, so the tunnel's shape is a careful compromise between hiding and breathing.

We'll see how that absorbing material works when we dive into the skin.

Everything warm gives off infrared light: you, a cup of tea, a car engine. The hotter it is, the more it shines. Night-vision and missile sensors are cameras for infrared.

Part of the air coming in through the intake flows around the outside of the engine instead of through it. That cooler air is blended with the hot exhaust in the duct, so the gas that leaves is far cooler than the gas inside the engine.

Imagine a round jet and a flat jet with the same amount of gas. The flat one has a much longer edge where it touches the cold air around it, and mixing happens at that edge.

The slot also hides the hot engine parts inside. A round nozzle lets a sensor behind the plane look straight up the pipe at the glowing turbine; a long, flat, curved duct doesn't.

The exhaust flows over this trough before leaving the plane. The tiles are made of materials that can take the heat again and again without cracking, a bit like the tiles on a space shuttle, but for much lower temperatures.

Sound matters too. Burying the engines and mixing the exhaust also makes the B-2 quieter than a normal jet of its size, so it's harder to hear coming.

Much of the B-2's structure is carbon-fibre composite, not aluminium: thin threads of carbon glued in plastic, layered in different directions. It's strong, light, and can be shaped in big smooth pieces with few joints.

The honeycomb core is like a sheet of tiny hexagonal paper cells glued between two skins. It makes the panel stiff without much weight, the same idea as corrugated cardboard.

The exact layers and materials are secret. What we show here is the general idea that radar-absorbing skins use, drawn thicker so you can see each layer.

Inside iron are trillions of tiny magnetic regions. A passing radar wave flips their direction back and forth billions of times a second. They can't quite keep up, and that 'lag' turns wave energy into heat.

The heat is tiny, far less than you'd notice. Radar waves carry very little energy by the time they reach a plane far away. The point isn't to warm the plane, just to leave nothing for an echo.

Materials like this are called radar-absorbent material, or RAM. Newer B-2 coatings are sprayed on like paint instead of laid on as tape, which makes them easier to repair.

Each pass through the absorber might cut the wave's strength by more than half. Going in and coming back out doubles that loss.

But the absorber has a second, cleverer trick. The small part that does come back out is timed to meet another echo and cancel it. Scroll on to see how.

Whenever a wave passes from one material into another, a little of it reflects at the boundary. That's why you see a faint reflection in a shop window even though most light goes through.

The two echoes are made from the same incoming wave, so they have exactly the same rhythm. The only difference is how far each one travelled.

This is called destructive interference. For a common radar wave 3 cm long, a quarter wave is 7.5 mm. Inside the absorber, waves travel slower and get shorter, so the real layer can be even thinner.

The cancelling only works perfectly for one wavelength. Designers stack layers or choose materials that work over a range of radar types, and the absorbing iron weakens the echo too, so the two tricks work together.

Noise-cancelling headphones use exactly the same idea with sound: a tiny speaker plays the opposite of the noise, and the two cancel in your ear.

Remember the 'fourth power' rule: cutting the echo 10,000 times shrinks a radar's detection range only 10 times. That's why every layer of trick matters; each one multiplies the others.

Stealth also has limits. Very long radar waves (metres long, like old early-warning radars) are less fooled by the shape and absorber, though they're too blurry to aim weapons. And no coating is perfect from every angle. Stealth buys time and distance, not magic invisibility.