What happens if a flat geometric shape, such as a rectangle, begins to rotate rapidly relative to one of its sides?
Only by rotation can we create a new geometric solid in space.
The side surfaces of the cylinder are formed by the sides of a rotating rectangle.
The official definition for such geometric bodies is as follows:
The solids of revolution are volume bodies that arise when a flat geometric figure bounded by a curve rotates around an axis lying in the same plane.
And here it is important that a flat geometric figure can be of a completely arbitrary shape.
For example, a curve that, when rotated, will form a vase or a light bulb. Such tools for creating bodies of revolution are very popular with those who work in 3D-design programs.
But from a mathematical point of view, the following geometric bodies of revolution are primarily of interest to us:
The cylinder is formed by a rectangle rotating around one of the sides.
The cone is formed by a rectangular triangle rotating around one of the legs.
A truncated cone is a part of the cone located between its base and the cutting plane parallel to the base.
Formed by rotating a rectangular trapezoid around its side, perpendicular to the bases of the trapezoid.
The ball is formed by a semicircle rotating around the diameter of the cut.
During the rotation of the contours of the figures, a surface of revolution arises (for example, a sphere formed by a circle), while during the rotation of filled contours there appear bodies (like a ball formed by a circle).
Ellipsoid is a surface in three-dimensional space, obtained by deforming a sphere along three mutually perpendicular axes.
The torus is formed by a circle rotating around a straight line that does not intersect it.
In the usual sense of the torus - it is a "bagel".
A paraboloid is a surface that is formed as a result of a rotation around the axis of a curve formed by a graph of a parabola. Hence the name parabol-oid.
A hyperboloid is a surface that is formed as a result of rotation around the axis of a curve formed by a graph of a hyperbola. Accordingly, the name is hyperb-o-loid.
How to make a cylinder of paper?
We suggest you build a cylinder model with the following characteristics: height 150 mm, base diameter 65 mm.
Download cylinder net
Truncated cylinder, a geometric body, cut off from the cylinder by a plane that is not parallel to the base.
We offer you to build a model of a truncated cylinder, with the following characteristics: height 165 mm, base diameter 65 mm, the angle of inclination of the section plane 45 °.
Download net of a truncated cylinder
Assembly scheme of a truncated cylinder:
1. Connect the net, so that you get a tube.
2. To avoid difficulties and shifts when gluing the bases, we recommend that you first glue the auxiliary elements. This will allow to hold them smoothly when gluing and will create the correct shape of the geometric structure.
3. We perform a similar procedure for the lower base.
An example of a truncated cylinder from architecture:
In 1989, the planetarium in the shape of a truncated cylinder was built in Copenhagen (Denmark).
How to make a cone of paper?
We suggest you build a truncated cone model with the following characteristics: height 185 mm, base diameter 90 mm.Download cone net
What can be the cross section at the cone?
Depending on the angle of inclination of the section plane to the base of the cone, four sections can be formed: a circle, an ellipse, a parabola, a hyperbola.
An example of architecture: the base of the Ostankino television tower in Moscow has the shape of a truncated cone:
The elliptical cone assembly scheme:
1. Connect the unfolding, so that you get a tube.
2. To avoid difficulties and shifts when gluing
bases, we recommend that you first glue the auxiliary
items. This will allow them to hold them smoothly.
when gluing and will create the correct shape of the geometric structure.
3. We perform a similar procedure for the lower base.
Note! Other rotation bodies cannot be made from paper. Only: cylinder and cone, including all possible sections.
It would seem, what else are we talking about here? It is impossible, therefore it is impossible. But there is a small "but." From paper, you can build a body close to such rotation bodies as a ball, ellipsoid, torus, paraboloid and hyperboloid.
That is, the more petals the sweep contains, the more the model will approach the rounded shape.
How to make a paraboloid out of paper?
We suggest you build a model of a paraboloid, with the following characteristics: height 130 mm, base diameter 85 mm. Download a paraboloid net.
Paraboloid assembly scheme:
1. Glue the upper part of the scan.
2. We connect together. Formed domed part.
3. To avoid difficulties and shifts when gluing the base,
we recommend that you first glue the auxiliary elements.
This will allow you to hold them smoothly when gluing and
will create the correct shape of the geometric design.
An example from architecture.
We couldn’t find the buildings that exactly repeated the parabola formula. But, nevertheless, in London (United Kingdom) there is a skyscraper with a very unusual shape.
The Mary Eks skyscraper, referred to by locals as “cucumber” (eng. The Gherkin), has no corners, which prevents the wind flow to flow down and provides natural ventilation. The height of the 41 storied building is 180 meters. The diameter of the building at the base is 49 meters, then the building expands smoothly, reaching a maximum diameter of 57 meters at the level of the 17th floor. Further, the design narrows, reaching a minimum diameter of 25 meters.
How to make a paper hyperboloid?
Hyperboloid assembly scheme:
1. Connect the two "smallest" strips through the center.
2. It should turn out as shown in the figure.
3. Glue two more strips, those that are medium in size.
5. Glue the largest strips.
7. Connect the scan together.
8. To avoid difficulties and shifts.
when gluing the bases,
An example of a hyperboloid from architecture.
There are a lot of buildings with a hyperboloid:
The very first constructions were created under the guidance of the Russian engineer Shukhov V.G. - the famous Shukhov Tower in Moscow, year of construction 1922.
Of the modern buildings, we note the tower in Guangzhou (China), an altitude of 600 meters, the year of construction 2010.
In addition, we found a very beautiful Tornado Tower in Doha (Qatar). 195-meter construction, erected in 2008, has its own indescribable style.
In order to compare the sizes of the resulting models of rotation bodies, we tried to assemble them on the same surface together with prisms from the Magic Edges No. 16 issue.
It turned out a whole mathematical city of paper that fits on the table!