tag:blogger.com,1999:blog-382677439644973005.post8344063494311740653..comments2023-09-24T17:08:13.142+02:00Comments on Physics intuitions: Geometric representations of higher degree binomialsArjen Dijksmanhttp://www.blogger.com/profile/09450431291713605237noreply@blogger.comBlogger11125tag:blogger.com,1999:blog-382677439644973005.post-17794720832307629552022-02-19T15:36:51.248+01:002022-02-19T15:36:51.248+01:00This comment has been removed by a blog administrator.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-382677439644973005.post-71599148433306408122014-06-06T15:33:02.291+02:002014-06-06T15:33:02.291+02:00@Shantanu. Yes of course we can draw 3D volumes fo...@Shantanu. Yes of course we can draw 3D volumes for (x+y)^n, n=4, 5, 6,... like we can draw 2D areas for those binomial expansions. We just have to divide the volumes into x/(x+y) and y/(x+y).Arjen Dijksmanhttps://www.blogger.com/profile/09450431291713605237noreply@blogger.comtag:blogger.com,1999:blog-382677439644973005.post-63495577843970936922014-06-06T08:15:54.693+02:002014-06-06T08:15:54.693+02:00can we draw geometry of (x+y)^n ; n>=4 ; as dra...can we draw geometry of (x+y)^n ; n>=4 ; as drawn for (x+y)^3 in fig-"Geometric proof of cubic binomial (original at The Geometry of the Binomial Theorem, Math Awareness Month site) " above????Anonymoushttps://www.blogger.com/profile/14648358346564487506noreply@blogger.comtag:blogger.com,1999:blog-382677439644973005.post-76883183552917776902011-10-19T22:50:00.797+02:002011-10-19T22:50:00.797+02:00Dijksman graphs? My parents would love it ;-)Dijksman graphs? My parents would love it ;-)Arjen Dijksmanhttps://www.blogger.com/profile/09450431291713605237noreply@blogger.comtag:blogger.com,1999:blog-382677439644973005.post-28662105156398659012011-10-19T18:52:25.894+02:002011-10-19T18:52:25.894+02:00"electrical charges "find" the shor..."electrical charges "find" the shortest path " should read "electrical charges "find" the shortest path through a submerged maze in a salt solution" - lifebiomedguruAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-382677439644973005.post-73785032155191573172011-10-19T18:51:09.659+02:002011-10-19T18:51:09.659+02:00Yes, viewing would be cool, and the application I ...Yes, viewing would be cool, and the application I suggest goes further; Also can imagine a video of problems being solved by interacting Dijksman graphs. The problems stated as equations have equivalent graphical representations; their relationships in Dijksman graph space can be studied, and their manipulations in Dijksman graph space would be very interesting to watch. Maybe some simple well-known elementary polynomial problems being solved would be nice to see (and they would suggest a set of consistent operations in this space). Some intractable problems could be handily dealt with and here is a conjecture: With a set of consistent rules for abstraction of operations in place in graph space, a solution rendered in Dijksman graph space is equivalent to its proof. NP-Completeness is bunk; electrical charges "find" the shortest path because they are at once an abstraction of the problem, and the solution. You're onto to something profound here, I think. We use graphs to SEE unsolvable problems, why not use your to SOLVE them. - lifebiomedguruAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-382677439644973005.post-83776402006540131282011-10-19T17:03:29.838+02:002011-10-19T17:03:29.838+02:00Yes, if someone could develop this way of viewing ...Yes, if someone could develop this way of viewing higher dimensions, it would but great. Maybe Wolfram indeed?Arjen Dijksmanhttps://www.blogger.com/profile/09450431291713605237noreply@blogger.comtag:blogger.com,1999:blog-382677439644973005.post-11346185520561422762011-10-19T13:29:50.868+02:002011-10-19T13:29:50.868+02:00"each" above should read "each othe..."each" above should read "each other".Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-382677439644973005.post-85036214262028102842011-10-19T13:29:02.284+02:002011-10-19T13:29:02.284+02:00There are those who say it's difficult for us ...There are those who say it's difficult for us to visualize dimensions n>4. I as driving home last night and had this same insight and my search for similar thought 'out there' brought me here. Actually, this representation of higher dimensions that result from higher degree binomials can be used to symbolically represent the hyperspaces and for any system where variables relate via addition/subtraction multiplication/division. Take the colorscale and make solutions < 0 one range; solutions > 0 another. I foresee applications for understanding the complexity of solution space if this is done algorithmically; undefined solutions painted black. moreover, a set a rules could be made such that solution graphs could be related to each and mathematical manipulations of solution spaces could be projected into and executed in this graph space; and then the mathematical solution could be read from the graph. Wolfram might be interested. -lifebiomedguruAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-382677439644973005.post-1033450194355070762010-12-29T22:24:41.503+01:002010-12-29T22:24:41.503+01:00I think you lost me:-) I don't exactly see how...I think you lost me:-) I don't exactly see how I have reduced the cubes and parallelepipeds to areas. I rather thought of x^3 as (x reduced by x reduced by x) for x<1, or (x increased by x increased by x) for x>1.Arjen Dijksmanhttps://www.blogger.com/profile/09450431291713605237noreply@blogger.comtag:blogger.com,1999:blog-382677439644973005.post-35612999700989063132010-12-29T15:30:19.053+01:002010-12-29T15:30:19.053+01:00It seems that you've projectivized the volumes...It seems that you've projectivized the volumes to create a sort of level curve graph in R^2. It's very nicely done and a very keen insight. There are so many visual subtleties in math.rcamp004http://www.twitter.com/rcamp004noreply@blogger.com