Different spirals follow.
Most of them are produced by formulas. Spirals by
Polar Equations
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(2) The motion with a constant angular velocity moves the point on a spiral at the same time. - There is a point every 8th turn. (3) A spiral as a curve comes, if you draw the point at every turn. You get formulas analogic to the circle equations. Circle
Spiral The radius r(t) and the angle t are proportional for the simpliest spiral, the spiral of Archimedes. Therefore the equation is: (3) Polar equation: r(t) = at [a is constant]. From this follows (2) Parameter form: x(t) = at cos(t), y(t) = at sin(t), (1) Central equation: x²+y² = a²[arc tan (y/x)]².
Fibonacci Spiral
The picture pair makes a 3D view possible.
Conical Helix top
You can make the conical helix with the Archimedean spiral or equiangular spiral. The picture pairs make 3D views possible.
Loxodrome, Spherical Helix
I suppose that you have to explain this effect in the same way as a bimetallic bar. You create a bimetallic bar by glueing together two strips, each made of a different metal. Once this bimetallic bar is heated, one metal strip expands more than the other causing the bar to bend. The reason that the strip of paper bends is not so much to do with the difference in temperature between the top and bottom side. The knife changes the structure of the surface of the paper. This side becomes 'shorter'. Incidentally, a strip of paper will bend slightly if you hold it in the heat of a candle flame.
Costume jewelleries also take spirals as motive.
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