![]() ![]() In addition to wanting to give his grandfather an answer, John was intrigued by his experimental result and wanted to know if it was exact, so he mentioned the problem to friends and colleagues around Harvard and would lend the model to anyone who would work on the problem. Given that John's expertise was chemistry, he promptly dunked it in a beaker of water and discovered its volume was quite close to (if not exactly equal to) (4π /30)r 3 i.e., one tenth the volume of one of the four spheres of which it is the intersection. Peterson figured his grandson might be able to do what he hadn't, so he gave John one of his models and asked him to figure out what the volume was. Maggio, became a chemistry grad student at Harvard. Of course, not knowing was no real obstruction, but over the years he wondered. He wondered how much lead he needed, and hence what was the volume of his spherical tetrahedron. Peterson cast lead models using forms made from old-fashioned toilet tank floats. This being the length of the line through midpoints on opposite edges.) (This latter usage might tempt you to expect that such a spherical tetrahedron is a figure of constant width r, but in fact at its widest it has width (√3-1/√2)r = 1.025r, It is also nowadays called a Reuleaux tetrahedron, in analogy with a Reuleaux triangle. Around that time, Peterson set himself the problem of constructing models of the 3 dimensional version, which (in analogy with circular triangles) he called a The company logo consisted of a pair of Reuleaux triangles (as still can be seen on the company web site the upper left of their home page has a globe with two Reuleaux triangles and below that the list of their products is superimposed on the profile of a Reuleaux triangle). Gearench Manufacturing Company, probably in the 20s or 30s (the company was founded in 1927). Arvid Peterson, an engineer working for a Texas oilfield tool works, The problem (as it came down to me) of finding the volume was first posed by J. The 3 dimensional analogue of the Reuleaux triangle is the intersection of the interiors of 4 spherical balls of radius r, each of which goes through the centers of the other three: The area of the equilateral triangle is √3/4 r 2, hence (60/360)π r 2, where r is the length of a side of the equilateral triangle. ![]() Is a circular sector on an angle of 60 degrees, hence has area The central equilateral triangle (shown in the next figure in red) together with one of the three circular wedges (shown in green) ![]() It's easy to compute the area of a Reuleaux triangle. Here is a page showing a rolling Reuleaux Triangle and here is one showing it rotating (and thus how a drill bit can be made to drill a nearly square hole). This has the property of being a curve of constant width. The Reuleaux triangle (the union of the red and green regions shown in the next figure) is the specific case in which the triangle is an equilateral triangle and each arc is centered at the opposite vertex. Hauterive Cistercian Abbey in Fribourg, Switzerland,Īnd here is a photo inside the Eglise Notre Dame in Bruges showing another example see the top of the windows in the background). A particularly special case known as a Reuleaux triangle (named after Franz Reuleaux, a German engineer from the 1800s, although the figure is much older, occurring in Gothic architecture for example Its volume: ((8/3)π - (27/4)cos -1(1/3)+(√2/4))r 3.Ī circular triangle is a plane figure resulting from a triangle whose sides have been replaced by circular arcs. ![]() With some exposition and history of the problem: Volume and Surface area of the Spherical Tetrahedron (AKA Reuleaux tetrahedron) by geometrical methods Date: January, 2010 (updated September 10, 2017) ![]()
0 Comments
Leave a Reply. |
Details
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |