source file: mills2.txt Date: Fri, 11 Jul 1997 10:21:16 +0200 Subject: CPSs with no factor of 1 From: "Paul H. Erlich" Actually, CPSs with no factor of 1 arise naturally in the construction of Euler genera from CPS building blocks. For example, the Euler genus 3.5.7 can be constructed from the CPSs 1)4[1,3,5,7], 2)4[1,3,5,7], and 3)4[1,3,5,7]. The intersection of the first two is the CPS 1)3[3,5,7], and the intersection of the last two is 2)3[3,5,7]. So CPSs without a factor of 1 are "where you put the glue" when constructing Euler genera. In the triangular lattice, the CPSs are symmetrical structures, while the Euler genera are skewed parallelopipeds. In fact, according to the triangular lattice philosophy, there are many other structures just as simple as Euler genera. For example, if you use 1)3 [1,3,5] and 2)3 [1,3,5] as the connecting faces for the first three CPS mentioned above, you get such a structure. (The individual CPSs will, in general, have to be transposed with respect to the raw multiplication results; the fact that you don't have to transpose them in the construction of the Euler genus is a mere mathematical curiosity.) More interesting is what happens when you use all possible connecting faces of a given CPS; at least in some cases, this results in what Erv Wilson called a "stellation." Or using all possible 1)n or (n-1))n CPSs connected to a given point; that results in a Partch diamond if the n factors are consecutive odd numbers starting with 1. . . Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 11 Jul 1997 10:22 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA26474; Fri, 11 Jul 1997 10:22:40 +0200 Date: Fri, 11 Jul 1997 10:22:40 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA26442 Received: (qmail 14868 invoked from network); 10 Jul 1997 21:22:35 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 10 Jul 1997 21:22:35 -0000 Message-Id: <970710171820_-2080181448@emout19.mail.aol.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu