Topological Shapes and More!

In this section, we would like to discuss how geometry helps inventors find new and unusual solutions. The most surprising solutions were brought to mankind by a division of geometry called "topology". Topology explores certain spatial properties of geometrical figures. These figures in space remain unaltered when the space is twisted, stretched or deformed in any way.

Let's start our inventive voyage into topology.

One of the most exciting topological shapes used in inventions is the "Mobius strip", named after a German mathematician, August Ferdinand Mobius (1790-1868). He first mentioned of this spectacular strip in literature was in 1858. But was he the discoverer? In a museum in France, visitors can find an exhibit of an ancient puzzle, the decoration of which contains a twisted strip with a continuous black line running along the length. Also in Paris, seamstresses, hiring new employees, gave them an impossible task: to stitch to a dress a lace with ends twisted 180 degrees. As it goes by the legend, Mobius mused about the properties of this strip when he noticed, on one of his servants, one of these incorrectly stitched lace.

While researching the materials for our timeline of inventions, we found a notice stating that most likely, most of the inventions on the timeline were reinvented after they first appeared in ancient times! 

You can make your own Mobius strip. It's very simple. Cut out a paper strip, twist the strip 180 degrees and glue the edges together. If you are in a skeptical mood and your imagination allows you, take a look at the picture of a Mobius strip. 

The Mobius strip possesses downright magical qualities. And the most interesting quality is that it is so simple to make one. Your creation (if you made one) has an interesting property; it has only one surface! Want proof? Try to color the outside with a color and the inside with a different color. What you should soon realize is that that task is impossible. Let's take a closer look at the qualities of this strip. If you think you know enough about the Mobius strip, click here to skip. For those of you are who are more curious, read on.

When it is cut in half along the diameter, instead of creating two new strips, the strip becomes twice as long. If a strip with ends twisted 360 degrees is cut along the diameter, then a chain of two strips is obtained, each of which is a one-surfaced.

If the one end is twisted 540 degrees before the ends are glued, and the strip is then cut along its diameter, the result is very unexpected: the strip doubles in length and on it appears a knot, which can be gotten rid off only by ripping the strip. 

Now what if we cut the strip not along the diameter, but say along 1/3 of the width? This time we get a chain of two rings: one that has 1/3 or the width of the original and 2 surfaces and another that is 2/3 of the width of the original and still only has one surface. We advice to try all of the above cuts by yourself. This will bring you in-hand experience and are enthusiastic about your possibilities of creating your own "Mobius" inventions. 

Due to its magical properties, this strip earned an honorable position in the inventive world.
Considering the age spectrum of our visitors, we will start the exhibition of "Mobius-based" inventions with an invention titled "Mobius Strip Puzzle", picture of which can be found above. On our website, you can learn how to find this patent by its number, given above, learn about the patent, and create your own puzzle to surprise your friends.

The next item in this exhibition is the "Mobius Railroad". An inventor named I. E. Burlack surprised his kids with an unusual railroad. He twisted a toy railroad track with magnetic rails into a Mobius strip and also made a train with magnetic wheels to go with it. The kids were amazed; their train twisted in space and ran upside down. Such path of twisting surfaces has industrial applications. For example, the path is effectively used in canting sheets of metal and other planar objects without human involvement. 

In our life, people mix many materials, from cocktail to concrete. The mixing process is very simple: the materials mixed must be turned over and inventors established that the Mobius strip can do this automatically, simply by its rotation. A brainy group of inventors were granted 4 patents for different designs of such mixers. More surprising is the "wind mill" in which the rotor is replaced with a Mobius strip. Now, this windmill does not depend on the direction of the wind.

As well as being used in toys and industrial applications, the one-surface property of the Mobius strip is used in electrical elements. For example, in the US patent 4599586, Thomas J. Brown proposed the following: "An electrical element, utilizing a capacitive enclosure of a Mobius strip and the spatial phenomenon thereof, to measure the voltage and phase differences of input signals or to act as a filter to attenuate current flow of a resonant frequency and the harmonics of that frequency while passing intermediate frequencies. This invention contemplates utilizing the Mobius resistor, which is known in the prior art to be two conductive surfaces separated by a dielectric material twisted 180 degrees and connected to form a Mobius strip to provide a resistor, which has no residual self-inductance.

All of the inventions above dealt with the "classic" Mobius strip, once twisted strip with connected edges. Can other one-surfaced objects exist. What do you think? Sure could. One of such strips was created by Josiah Manning and you can examine his version in the May 2000 issue of Scientific American.

Courtesy of Abram Teplitskiy

Let's take a look at another one-surfaced body. This time, it's a spatial object bearing the name of the "Klein bottle". Anyone can create their own Klein bottle by "following the following":

Obtain a funnel-shaped pipe. Take the narrow end, insert it into the pipe near the wide end and pull it, through the inside, all the way towards the wide end (the narrow end should be inside the wide one). Now glue the ends together. If an ant is placed inside this bottle, then it, without crossing one end will explore the outside and the inside of a fantastic shape. (as described in a book "By Azimuth of Technical Creativity" [in Russian] by Abram Teplitskiy et. al.)

Surprisingly, the properties of this bottle have not yet been technically taken advantage of. You have all the possibilities to set foot in this unexplored region and make a fortune, hopefully.

Try! You can be the first in the world!

It may seem as if though you were just introduced to the wildest part of geometry, but it isn't. Now let's learn how to drill a square-shaped hole using a triangle-shaped drill.


US Patent #4074778

This problem can be solved by using the so-called "Reuleaux triangle". The general idea of this triangle is shown here . It combines the advantages of an equilateral triangle and a circle. If it is rotated "eccentrically", its sides will trace the shape of a square. Such a drill has been patented with the title "Square Hole Drill" (US Patent 4,074,778). The drive train of this particular drill operates by two counter rotation of the center axis of the planet gear about itself as it meshes with and rotates within a larger fixed ring gear. The second rotation which is counter to the first is also caused by the planet gear as it meshes with the ring gear and is the circle formed by the center of the planet gear as it rotates inside the ring gear.

The Reuleaux triangle can bring profit working as a part of engines (for example, US Patent 3,922,120). The authors of this patent propose a rotary engine having a housing with a four-lobed cavity therein and three-lobed rotor disposed in the cavity, the cavity and the rotor having cooperating surfaces all of which are sections of tight circular cylinders and in which the rotor moves within the cavity by sequential pivotal movement of the rotor about its lobe apexes in the coaxial with the cavity is coupled to the rotor by means allowing the rotor axis to move towards and away from the axis of the cavity as the rotor rotates within the cavity.

Although the description above may look complicated, anyone can design and build such a triangle. Start obtaining a piece of paper, a ruler, and a compass. Next, sketch an equilateral triangle and label points a, b, and c. Put the sharp point of your compass on point a and with the pencil-point draw an arc from b to c. Repeat the process by putting the sharp point on point b and drawing an arc from c to a and then make the last arc connecting c and b. If you circumscribe a square around this triangle, the rotation of the triangle will completely trace the boundaries of the square. The area not covered by the rotation of the triangle around the center, point O, is about 10 times less than the area not covered by the rotation of an inscribed circle. Now, draw a point O1 a certain distance from point O and try rotating your triangle around your new center (O1 will rotate around O). The corners of the triangle will "visit" the corners of the square and cover practically all of the square's area. This principle allows the drilling of square-shaped holes in soil, metal and other materials.

Now take a second to think if you have any interesting solutions to problems with the use of the Reuleaux triangle… If you don't, place it in your head and maybe soon you will find a problem that can be solved with this triangle. Remember, a problem can and usually does have more than one solution, so collect all kinds of knowledge in your memory. We understand that all knowledge can't be stored in one's brain, and it doesn't have to be. Take another second and make notes of interesting objects discussed here or anywhere else. Start your own "bank of ideas"! Your bank of ideas can be in any format, any surface, even on topologic shapes!

We began with near magical math applications in practice and now, you will see the simplest math objects can perform magical jobs.

All of the objects described above have been unusual shapes. Now let's take a look at simple, well-known geometrical shapes that can also perform magical tasks.

Since ancient times, people attempted to explain why objects as big as planets and as small as a drop of dew and bubbles of air in water are the same shape. The prominent scientist Galileo spent many years examining a drop of dew and explain why it's spherical. Now you can consider yourself as smart as Galileo, because you know that spheres, when compared to other shapes, have the highest volume. By definition, sphere is a set of all point in space equidistant from a center point. You can refresh your memory of sphere's properties by looking in your geometry textbook. We will only remind you of one, the most important in our opinion, property of the sphere: with a given volume, the sphere has the smallest surface area of all possible geometrical figures. This is why the first spacecraft, Sputnik, drops of dew and air bubbles are spherically shaped.

Let's take a look at some technical applications of properties of the sphere. One of the simplest uses of spheres is in separators for division of particles by size. Usually, sieves are used for this purpose, but if we are dealing with magnetic materials, then sieves are ineffective. Inventors proposed creating a new type of sieves, one that used spheres. When spheres of the same size are positioned together on a flat surface, holes of identical size appear and as a result, a perfect sieve is obtained. One of the advantages of this type of sieve, is the ease of cleaning: all it takes is taking the spheres apart.

Another interesting use of spheres was patented in the United States, by Tsuchiya Shozo (US Patent 3,716,093) One of the requirements of vehicles, in order to travel in a stable state over road surfaces, is the lowest possible center of gravity and a moderate clearance.
The wheel comprises a wheel rim, a pneumatic tire attached to said rim and a plurality of balls partly filling the cavity of the pneumatic tire. Preferably, the balls may be solid, and consist of metals having a relatively high density such as steel, lead or lead-containing alloys, and are generally about 1 to 3 cm in diameter. It is desired that the total weight of these balls range from a minimum of substantially increasing the weight of a wheel as a whole to such a maximum that when the axis of the wheel is brought to a substantially horizontal position, the upper surface of the packed layer of balls touches the top of the round cross section of that part of an annular tire cavity which successively faces the ground.

Where a wheel containing such balls is fitted to the axle, the weight of balls acts on the inside of the bottom of the tire cavity facing the ground, bringing the center of gravity of the wheel to a far lower point than the axial center of the axle. When the wheel rotates, the balls themselves also roll and always tend to move towards the lowest position in the tire cavity, so that a low center of gravity of the wheel is maintained even while it is rotating.

Do you agree that if we take a plurality of tiny spheres, they act as a form of liquid? Examine this analogy closer: these spheres take the shape of the container they are in, which is one of the main properties of liquid, and they retain a horizontal surface, as does water.

Courtesy of
Grolier Educational

Now let's take a trip in time and visit our prominent friend, Archimedes. He not only invented the screw, which in his time allowed irrigation of land, which was a humongous problem, but also made a plethora of other discoveries, one of which was a method of measuring volume of eccentric shapes (shapes other than a cube, or another shape whose volume can be measured through volumes). By his method, an object is

Courtesy of Grolier Educational

submerged into a scaled container; the increase in the level of water is the volume of the object. Civil engineers often face a problem of measuring the volume of porous and/or dissolvable materials: the water will flow into the porous cavities and affect the result and/or dissolve the material.

The solution was found in the analogy mentioned above. Instead of water, tiny spheres were used. Like water, they contoured the object tightly and provided accurate results, yet they did not dissolve the object or flow into the tiny porous cavities.

While assembling this section, we felt as if someone was closely watching us. The feeling would not disappear. We started scrolling this page up and down until we noticed that it was Archimedes! He silently asked: "I have contributed a great deal to mankind. The screw, for example! You can see how it served people on the left. And what have you done?"

Being responsible webmasters, we convert Archimedes' silent message to you: become an inventor!

 

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