Date: 11/21/2001 at 13:10:49 From: Russell Kopp Subject: Geometry Hi: I have a kitchen table whose functionality involves the use of math. First I'll explain the table: From a crow's eye view, in its folded state the table is a rectangle, with adjacent sides of different length. (I suppose it could have been square, but I'm not sure). The tabletop has hinges on one of its longer sides - for reference I'll call this its east side. Attached to these hinges is another tabletop of equal size that sits on top of the original tabletop when the table is in its smaller, folded state. The bottom of the two tabletops is connected on its bottom or under side to a pivoting device that allows the bottom tabletop (and of course the top tabletop attached to it) to be rotated clockwise 90 degrees. Once this rotation is effected, the top of the two tabletops may be unfolded from the bottom tabletop, revealing a table twice as large in size as the original. In its unfolded state, the overlap of the larger table top on its base is such that north and south table ends overlap the base by equal amounts, and the west and east ends overlap by equal amounts as well. The overlap of adjacent sides need not be the same. Were the table split into four quadrants, the pivot would clearly lie in the upper right (northeast) quadrant. I wish to know how to determine the correct pivot point such that it meets all the guidelines discussed above. I should note that the table's actual dimensions in its folded state are 35.4375" by 24.4375". I've done a little research, and should note for those with geometry backgrounds that this is not a golden rectangle (a rectangle whose adjacent sides are in a ratio of approximately 1:1.618.) I say this because I thought a golden spiral's limit might be the pivot point - though it appears not to be the correct point. Thanks, Russ
Date: 11/21/2001 at 20:20:44 From: Doctor Rick Subject: Re: Geometry Hi, Russ, thanks for writing to Ask Dr. Math. I like questions like this - discovering math in everyday things! I've seen the kind of table you're talking about, but I hadn't asked myself the question you have. It turns out that the answer is simpler than you thought. Here is a figure: +---------------+ | | ..........|...............|.......... : | | : : | | : : | | : : | | : : | | : : | | : : | | : ..........|.......B.......A.......... : | | : : | | : : | | : : | | : : | | : : | | : : | | : ..........|...............|.......... | | +---------------+ You have stated that the opened table is centered on the base; you didn't state, but I think I can assume it, that the folded table is also centered on the base. Thus I've drawn the folded (solid lines) and opened (dotted lines) tabletops centered at the same point. The point marked A, in the center of the hinged side of the folded table, must move to point B, the center of the entire table, when the top is rotated 90 degrees. All we need now, to find the center around which the tabletop is rotated, is to locate a point such that lines drawn from it to A and B are the same length and form a 90-degree angle. That's easy: +---------------+ | | ..........|...............|.......... : | | : : | | : : | | : : | | : : | | : : | O | : : | / \ | : ..........|.......B.......A.......... : | | : : | | : : | | : : | | : : | | : : | | : : | | : ..........|...............|.......... | | +---------------+ We have here a right isosceles triangle OAB; OA and OB are equal, so A can rotate to B, and AOB is a right angle, so the tabletop is rotated 90 degrees. If the width of the folded table (perpendicular to the hinged side) is W, then the pivot point O is W/2 to the east of the center, and W/2 north. The proportions of the table are irrelevant. We just need a base that is at least half the width of the folded table. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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