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Starting with an isosceles right-angled triangle of equal sides of, say, 10cm, the length of the hypotenuse is sqrt(10^2+10^2)=sqrt(2*10^2)=10sqrt(2), using Pythagoras' theorem c^2=a^2+b^2. Now building on the hypotenuse as one side of a new right-angled triangle, we make the other side 10cm. So all we're doing is using c as a new b and finding the next value for c. This time the hypotenuse is sqrt(2*10^2+10^2)=sqrt(3*10^2)=10sqrt(3). Then the process is repeated, and the hypotenuse of the next triangle is 10sqrt(4)=20cm. Each segment of the spiral is 10cm long, but the radius increases as the square root of the natural numbers: 1, 2, 3, 4, etc. so the nth radius is 10sqrt(n). The length of the spiral is 10n. The spiral is usually shown mathematically as just the unit side, so the perimeter or length of the spiral is just n and radius sqrt(n). It's not a true spiral, because the edges are straight, whereas a true mathematical spiral is a continuous curve. The angle between consecutive radii is tan^-1(1/sqrt(n)) so it depends on where you are on the spiral. The largest angle is the first one at 45 degrees. The next is tan-1(0.7071)=35.3 degrees approximately, then 30 degrees, then 26.6, 24.1, 22.2, 20.7, 19.5, 18.4, 17.5,...

When we reach sqrt(16)=4, the sum of the angles comes to about 351.15 degrees, and the next segment of the spiral takes us a little past a complete revolution at 364.78 degrees. The project normally stops at this point, and the final result looks like the shell of a nautilus mollusc (which is rounded).

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