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Nets solids of revolution

solids of revolutionWhat happens if a flat geometric shape, such as a rectangle, begins to rotate rapidly relative to one of its sides?
We create a new geometric solid in space by rotation.

The sides of a rotating rectangle form the side surfaces of the cylinder.
get rotation cylinder
The official definition for such geometric solids is as follows:

The solids of revolution are volume solids that arise when a flat geometric figure bounded by a curve rotates around an axis lying in the same plane. Here it is important that a flat geometric figure can be a completely arbitrary shape.

For example, a curve that, when rotated, forms a vase or a light bulb. Such tools for creating bodies of revolution are very popular with those who work in 3D-design programs. solids created by rotation curves But from a mathematical point of view, the following geometric solids of revolution are of primary interest to us:

 

cylinder rotation
The cylinder is formed by a rectangle rotating around one of the sides.
rotation cone
The cone is formed by a rectangular triangle rotating around one of the sides.
truncated cone rotation
A truncated cone is part of the cone located between its base and the cutting plane parallel to the base. It is formed by rotating a rectangular trapezoid around its side, perpendicular to the trapezoid bases.
ball rotation
The ball is formed by a semicircle rotating around the diameter of the cut.
During the rotation of the figures' contours, a surface of revolution arises (for example, a sphere formed by a circle). During the rotation of filled contours, bodies appear (like a ball formed by a circle).
rotation ellipse
An ellipsoid is a surface in three-dimensional space, obtained by deforming a sphere along three mutually perpendicular axes.
torus rotation
The torus is formed by a circle rotating around a straight line that does not intersect it.
The usual sense of the torus is a "bagel".
paraboloid rotation
A paraboloid is a surface formed due to a rotation around the axis of a curve formed by a parabola graph. Hence the name paraboloid.
hyperboloid rotation
A hyperboloid is a surface formed due to rotation around the axis of a curve formed by a hyperbola graph. Accordingly, the name is a hyperboloid.

How to make a cylinder of paper?
How to make a cone of paper?
How to make a paraboloid of paper?
How to make a paper hyperboloid?
How to make a paper torus?

To compare the resulting models' sizes, we tried to assemble them on the same surface together with prisms from the Magic Edges #16 issue.

solids of revolution prisms
It turned into a whole mathematical city of paper that fits on the table!


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