tag:blogger.com,1999:blog-8438916308834541294.post6726597609562107319..comments2024-03-22T00:17:02.004-07:00Comments on Office chair philosophy: Generalised definition for negative dimensional geometryTGladhttp://www.blogger.com/profile/01082123555974465066noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-8438916308834541294.post-79792590213882777112021-09-30T23:23:27.410-07:002021-09-30T23:23:27.410-07:00This comment has been removed by the author.Unknownhttps://www.blogger.com/profile/05124790701809785568noreply@blogger.comtag:blogger.com,1999:blog-8438916308834541294.post-38918738299099996772017-10-23T03:09:24.904-07:002017-10-23T03:09:24.904-07:00Hi Claude thanks for the link. For reference, the ...Hi Claude thanks for the link. For reference, the properties are: monotonicity, stability, countable stability, geometric invariance, Lipschitz invariance, countable sets, open sets and smooth manifolds. <br /><br />The positive dimension D+ satisfies all those properties, and the negative dimension D- satisfies a negative version of them, e.g. if E \in F then dim E >= F. The combined 'signed dimension' does not satisfy all of those properties, but in defense of this:<br />1. a dimension satisfying these properties could never be negative, so it is a requirement of a real-valued dimension that you need to have different desirable properties. <br />2. I mention in the final sentence that the signed-dimension can be left in its component form, e.g. this shape is 1-1.26D and so the single signed value -0.26D is just an abbreviation. But each component individually does satisfy all the dimension properties you cited.<br />3. The single signed dimension does retain what I think is the most important property: the dimension of the Minkowski sum <+> of the sets equals the sum of the dimension of the sets. i.e. Dim(a <+> b) = Dim(a) + Dim(b)TGladhttps://www.blogger.com/profile/01082123555974465066noreply@blogger.comtag:blogger.com,1999:blog-8438916308834541294.post-28239507648601929762017-09-21T09:47:40.326-07:002017-09-21T09:47:40.326-07:00I don't like this negative dimension stuff, pa...I don't like this negative dimension stuff, partly because it breaks monotonicity: if E \subset F then dim E \lessorequal dim F. See Falconer's "Fractal Geometry" 2nd Edition p41 for further desirable properties of dimensions.claudehttps://www.blogger.com/profile/15781861105387935084noreply@blogger.com