tag:blogger.com,1999:blog-28172905.post349680867850674838..comments2024-03-03T17:53:46.947-05:00Comments on (x, why?): Lines in Planes(x, why?)http://www.blogger.com/profile/17499160002806879025noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-28172905.post-37390677042080825312014-08-20T12:05:52.205-04:002014-08-20T12:05:52.205-04:00in the pure world devoid of shoeboxes, yes, a Eucl...in the pure world devoid of shoeboxes, yes, a Euclidean line is the unique intersection of ANY two of the infinite number of planes including the line. <br /><br />Using consistent terminology (face, edge, vertex) in the realm of finite solids and (plane, line, point) in the realm of infinite space, bridging with 'face embedded in a plane', 'edge embedded in a line', may be of pedagogic value? <br /><br />Why my pedantic pedagogy? Skewness is only interesting for the infinite line, since non-intersecting non-parallel edges aka line segments might be either collinear or intersecting in extension when their embedding in infinite rays or lines are considered.<br /><br />All quibbling aside, using the shoebox's several pairs of non-intersecting edges as a familiar manifestation of Skewness is a good thing. It does make uniqueness and perpendicular-ness of the minimum approach appear obvious ... while proving those facts is more interesting. :-)<br /><br />Fun combinatorial problem <br />How many pairs of skew lines are defined by the (extensions of the) 12 edges of the shoebox or a cube ? <br /><br />I partition all pairings into intersecting, parallel, or skew (lema: that it IS a partition!) and so get <br /><br />comb(12,2)<br /> - 8*comb(3,2) <br /> - 3*comb(4,2) ? <br />= 24 = 2*E = 2^3 * 3^1<br />where<br />12,8 are from cube's F,E,V=6,12,8; <br />one 3 is E-per-V (and only because it's a cube the number of dimensions), other 3 is number of dimensions = # (orthogonal) pencils of parallel edges.<br />and 4 is # edges per pencil, <br />2 is number of lines in a pair whether skew, intersecting or parallel.<br /><br />But what is the direct counting calculation ? 12*(2+2)/2. Bill Rickernoreply@blogger.comtag:blogger.com,1999:blog-28172905.post-84635062392485691552014-08-18T20:50:47.574-04:002014-08-18T20:50:47.574-04:00Would it have been better, leaving the illustratio...Would it have been better, leaving the illustration aside, had I said that the edges were formed by two planes, or by the intersections of said planes?(x, why?)https://www.blogger.com/profile/17499160002806879025noreply@blogger.comtag:blogger.com,1999:blog-28172905.post-64068698523164382662014-08-18T15:55:27.019-04:002014-08-18T15:55:27.019-04:00"They each join two planes,"
<quibbl..."They each join two planes,"<br /><br /><quibble> <br /><br />They each join an <i>infinite</i> number of (infinite) planes, two of which are realized (or distinguished) as (orthogonal) faces of the shoebox.<br /><br /></quibble>Bill Rhttp://m.twitter.com/n1vuxnoreply@blogger.com