Huwebes, Pebrero 7, 2008

REULEAUX TRIANGLE: ROLLING TRIANGLES

Man is a rational being. Through man’s existence on earth he has been longing and striving for knowledge. He keeps on searching for the things that are good for the body, things that are good for the soul and things that are good for the mind. Man’s hope lies in his rational nature and his determination and courage to reach the success, therefore his quest for the “impossible dream” must go on, and he must continue fighting and searching for the things that are good.

The desire of man is simply to satisfy his needs. Thus, the ability of man to think mathematically existed. According to some mathematicians, everything on earth that man perceives can be interpreted using mathematical expressions. Meaning, these things can be explained through numbers and figures. Mathematicians believe that mathematics was related to numbers and figures. Through this, another discipline arose and it is called “Geometry”. Geometry was defined as the branch of mathematics that treat space and its relation, especially as shown in the properties and measurement of points, lines, angles, surfaces and solids.

Studies concerning the relationship of figures trigger my curiosity which was inspired by different theories. Apparently, Barbier’s theorem indicates that some figures aside from circle possess a constant width. This statement caught my concern pertaining to this figure called “reuleaux polygons”. Barbier theorem states that all shapes of constant width D have the same perimeter pi D. The width of a convex figure in a certain direction is the distance between two supporting lines perpendicular to that direction. A straight line is called supporting a convex figure if they have at least one common point and the figure lies in one side from the line. In any direction there are two supporting lines. Shapes of constant width are convex figures that have the same width in any direction. The circle has this property. Barbier theorem also states that all curves of constant width of width w have the same perimeter.

For reuleaux triangle, we need to construct an equilateral triangle. On each vertex, center a compass, and draw an arc the short distance between the other two vertices. The perimeter will be three nonconcentric arcs. This is a reuleaux triangle. It is not a circle, but, like a circle, it has constant width, no matter how it is oriented. It is not difficult to see this property, but you should prove it.

Kunkel, P. (1997) made some sketches of the reuleaux and discovered some other interesting properties. It can roll uphill, in a manner of speaking. Notice that as it rolls, its height is constant, but the height of its centroid changes. If it had mass, the centroid would be the center of mass. Imagine that it is standing on one vertex, so that the centroid is at its highest, and imagine that the surface is very slightly inclined. If it moves forward one sixth turn, the centroid will fall. So although the surface rises, the reuleaux is actually falling.

With this, it triggers my imagination..is it possible to use this type of polygons in car wheels??

3 komento:

  1. Mahina talaga ako sa Math! Hahaha! Kapag may tag na ipapakita ang transcript of records, matutuklasan ng mga taga-blogospere na napakabobo ko sa Math. Pero hindi ko pa rin siya maiiwasan lalo na noong nagpa-practice pa ako ng profession ko sa Phils. Buti ngayon dito sa Au, simple arithmetic lang. o",) Bibilangin lang kung ilan ang napulot na patay na mga manok! Hahaha!

    Reuleaux triangles and circles! Naalala ko sa hematology (study of blood) may tinatawag ring rouleaux formation! Magkaiba ang baybay pero magkasingtunog, 'ata?

    Sa rouleaux formation ang mga red blood cells nagsasapaw-sapaw, they're like stacked coins! Abnormal ito, ibig sabihin may problema sa kalusugan ang isang hayop. U

    Ako ang unang magko-comment dito kaya lubus-lubusin ko na. (,"o

    Sa tingin ko pwedeng maging gulong ng sasakyan ang REULEAUX triangle 'Yet. Baka dyan mas lalong maging tanyag si Supergulaman- ang imbentor ng bayan! o",)

    TumugonBurahin
  2. Genial fill someone in on and this mail helped me alot in my college assignement. Thanks you for your information.

    TumugonBurahin
  3. Sorry for my bad english. Thank you so much for your good post. Your post helped me in my college assignment, If you can provide me more details please email me.

    TumugonBurahin

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