May 2020 - mathfun@mailman.xmission.com

Re: [math-fun] Can an equilateral toroidal polyhedron
by Dan Asimov 23 May '20

23 May '20

Yes, nice! That's a particularly beautiful torus in the regular 6-simplex living in 6-dimensional space R^6. Call this torus T_7. Its vertices and edges form the complete graph K_7, and its triangles consist of exactly 2/5 of the possible 35 triangles made of 3 vertices. As a combinatorial torus, it is dual to the 7-hexagon (hexagonal) torus, whose hexagons form a maximally symmetrical 7-coloring of the torus. This shows that T_7 with its intrinsic metric (distances measured along its 14 triangles, not across 6-space) is also a hexagonal torus: One obtained by identifying opposite edges of a regular hexagon. Which suggests the question: Can the 7-hexagon hexagonal torus itself be embedded in R^6 so that each hexagon is a bona fide planar regular hexagon? —Dan Adam Goucher wrote: --------- For n >= 6, you can take a Császár polyhedron sharing the vertex-set (and indeed edge-set) with a regular 7-vertex simplex. The problems for n in {3, 4, 5} are certainly interesting! I wrote: ------- RWG wrote: ----- Subject: Re: [math-fun] Can an equilateral toroidal polyhedron have fewer than 32 faces? —rwg ----- If it's not embedded anywhere, it can have just two faces. If it's embedded in n-space for fixed n, this is a terrific question! (Likewise for other surfaces, like the Klein bottle.) ------- ---------

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Re: [math-fun] Can an equilateral toroidal polyhedron
by Dan Asimov 23 May '20

23 May '20

----- have fewer than 32 faces? —rwg ----- If it's not embedded anywhere, it can have just two faces. If it's embedded in n-space for fixed n, this is a terrific question! (Likewise for other surfaces, like the Klein bottle.) —Dan

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Re: [math-fun] Can an equilateral toroidal polyhedron
by Dan Asimov 23 May '20

23 May '20

I wrote: ----- In fact there are some triangulated tori embedded in 3-space with only 14 faces, known as Cszásár tori. ----- Correction: ----- In fact there are some triangulated tori embedded in 3-space with only 14 faces, known as Császár tori. ----- —Dan

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Re: [math-fun] Can an equilateral toroidal polyhedron
by Dan Asimov 23 May '20

23 May '20

Fred Lunnon wrote: ----- https://en.wikipedia.org/wiki/Toroidal_polyhedron claims "Stewart toroids" with 32 and 30 faces resp. --- though it's not immediately obvious whether the faces are indeed regular! ----- No, it's not at all obvious (that the triangles can be made equilateral)! In fact there are some triangulated tori embedded in 3-space with only 14 faces, known as Cszásár tori. They look very weird! In fact, there exist triangulated tori in 3-space with 2n faces for all 2n ≥ 14. —Dan

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[math-fun] Can an equilateral toroidal polyhedron
by Bill Gosper 23 May '20

23 May '20

have fewer than 32 faces? —rwg

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Re: [math-fun] Covid19 R_e considered misguided
by Henry Baker 22 May '20

22 May '20

Just looking at your picture "Daily New Cases in New York". Hmmm.... What the heck is that 7-day period doing in there? Clearly there is a significant amount of measurement bias, unless there is some "forcing function" having to do with peoples' weekly schedules -- e.g., religious services, weekend activities, etc. There have been suggestions that religious services have contributed to superspreading events. ---- I have a far bigger problem with this type of chart: my mental model has 2 dimensions (probability density as a function of time), rather than simply value as a function of time. My mental model has an entire probability distribution (PDF) for each moment (day) in time, so any particular chart such as the one you linked is merely a random walk through this 2-dimensional chart. This process is analogous to computing the quantum wave function and then "collapsing the wave function" to a particular set of observations. With these fat-tailed distributions, however, the distributions are so wide that they are extremely *shallow* in depth so as to sum to 1. As a result, any particular chart labeled "Daily New Cases in New York" sampled from this 2-D density v. time chart is going to have a vanishingly small probability by itself, so there are a gigantic number of charts that are essentially equally likely. How does nature choose one of these charts? How does anyone choose which one of these possible charts to believe? How do you attempt to fit such a 1-D chart onto this 2-D model, and how could you estimate the parameters, and how would you define "fit" ? At 11:01 AM 5/22/2020, Brad Klee wrote: >Ha ha, tired of "proof by # of Twitter followers", me too! Glenn Greenwald >published via The Intercept a nice essay about this "pathology" on May 18. > >As for Kermack-McKendrick, I am personally interested in using differential >equations to create functions that fit data. So, naturally I would ask, how >closely can SIR describe, for example, NYC cases: > >https://0x0.st/ipso.png > >This data set has exponential rise, exponential tail, and probably sharp >enough peak, so it looks like a SIR fit might work. That could be a challenge >for you. If you want to KILL SIR, why don't you just show that it doesn't do a >good job of fitting this data set? > >I will look at your analysis if you come up with something by the numbers. > >Cheers, > >Brad > >On Fri, May 22, 2020 at 12:07 PM Henry Baker <hbaker1(a)pipeline.com> wrote: >> I was obviously trying to be too cute; I'm tired of >> being told "you have to believe me because 400 of my >> closest friends have referenced my papers" aka "proof >> by # of Twitter followers" when my own lying eyes (or >> my own algebraic calculations) tell me something quite >> different. >> [...] >> At the risk of mixing metaphors, I'd like to drive >> Dr. John Snow's wooden pump handle through the heart >> of the Kermack-Mckendrick 'R0' differential equation >> models.

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Re: [math-fun] Covid19 R_e considered misguided
by Henry Baker 22 May '20

22 May '20

I was obviously trying to be too cute; I'm tired of being told "you have to believe me because 400 of my closest friends have referenced my papers" aka "proof by # of Twitter followers" when my own lying eyes (or my own algebraic calculations) tell me something quite different. An epidemiologist who doesn't understand "fat tailed" distributions is a contradiction in terms, since almost no distribution in epidemiology has a thin tail. So it may be time for these people to get better educated, retire, or at least shut up. Their misunderstandings are wasting months of billions of people's lives and trillions of dollars. At the risk of mixing metaphors, I'd like to drive Dr. John Snow's wooden pump handle through the heart of the Kermack-Mckendrick 'R0' differential equation models. At 09:47 AM 5/22/2020, Brad Klee wrote: >H.B.: Perhaps those on this list will agree that proofs should end with >'QED', not 'PhD'. > >Me: The current PhD system does not inspire too much confidence, but I'm >not sure how much better it is to trade 'PhD' for 'QED', especially in messy >circumstances. Anyways most people who write QED have a PhD, so what >are you really saying? We need experts with training in formal mathematics? > >The other issue is that QED is a Latin abbreviation, so it is very exclusive to >elite westerners. Someone like Madhava would never think of ending an >idea-in-verse with QED. Is this a reason for taking away credit from him? >Elite westerners unequivocally answer yes. Is that answer fair?

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[math-fun] One issue with SIR predictions
by Brad Klee 22 May '20

22 May '20

Figure ... : https://0x0.st/i9C7.png Due to covariance with scale parameters it is difficult or impossible to meaningfully fit exponential rise. If you fix the shape, you can estimate duration, or vice versa. In a worst case scenario, the model gains predictive power after the peak, as is shown by comparison of two SIR curves in the figure above. A major problem we could be seeing now is whether or not the distribution has a plateau. If it has a plateau, then the naive SIR is pretty much out the window, along with its particular parameters. Trying to fit SIR during a plateau is not a good idea. --Brad

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Re: [math-fun] Covid19 R_e considered misguided
by Henry Baker 22 May '20

22 May '20

Once again, the use of 'average' in the phrase 'average reproduction number' presumes that the numerical value of 'average reproduction number' has some relevance to something or other in the real world. If such a numerical value can vary from 0.9-100, that fact would belie any commonsense notion of 'average' -- meaning 'ordinary' or 'typical'. But wait, it gets much worse! This numerical value R0_whatever-you-want-to-call-it is used as the *base* for an exponential function ! I.e., we're looking at an exceedingly fat-tailed distribution of numbers (R0)^some-exponent, so any hope of 'fitting' data to such a mess is out of the question. But models that can't fit data, can't predict anything. Models that cannot predict anything are no longer science, but quackery aka witchcraft. Is it any wonder that none of these 'models' can predict even an order of magnitude of infection, much less even 1 significant (decimal!) digit? The next article I read that quotes some PhD 'expert' which refers to this R0 nonsense will cause me to vomit. Perhaps those on this list will agree that proofs should end with 'QED', not 'PhD'. At 08:30 AM 5/22/2020, Michael Kleber wrote: >Just for clarity of terminology: > >The epidemiological definition of R0 is the average reproduction number *in a completely susceptible population*.

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Re: [math-fun] Covid19 R_e considered misguided
by Henry Baker 22 May '20

22 May '20

Perhaps the first order of business is to strangle the old differential equation model(s) in the cradle, and start using *Monte Carlo* simulations of network-type models. Given the ubiquity of computing power these days, there's no excuse for restricting the search for the "car keys" only near the differential equation "lamp-post". At 11:18 AM 4/11/2020, Fred Lunnon wrote: > No doubt you're right about all this --- but any sharpening requires >further assumptions about the distribution of R_e among the population, >for which any credible model is presumably currently unavailable. WFL > >On 4/11/20, Henry Baker <hbaker1(a)pipeline.com> wrote: >> I've been reading about Covid19 and listening to a number >> of professional podcasts about Covid9 -- e.g., "This Week >> in Virology", which amazingly enough, is entertaining >> enough not to be a soporific. >> >> Reading between the lines, I'm coming to the conclusion >> that the concept of "R_e" ("effective" R), may be fatally >> flawed, as it tries to capture some sort of "mean" or >> "average" R in an inherently exponential setting. >> >> Thus, if one person has an R_e of 0.9 and another person >> has an R_e of 40.0, there isn't a good way to average the >> two R_e's to compute a composite R_e. What if a vanishingly >> small fraction of infected people induce the vast majority >> of cases? We know that spreading is a *network* phenomenon >> which means that it is highly likely to have a *fat tail* >> distribution. >> >> If I'm correct about this, then the standard epidemic >> model is also fatally flawed, and hence unreliable for >> making trillion-dollar decisions. What won a Nobel >> prize a century ago is today an undergraduate homework >> exercise. We desperately need a better class of models >> for today's pandemics. >> >> For example, in Massachusetts in the early going, almost >> 100% of the confirmed cases stemmed from a single meeting >> of a single company in a single downtown Boston hotel. >> >> As another example, in Chicago, most of the early cases >> stemmed from a single funeral and a single birthday >> party. >> >> So if it is true that there are "super spreaders", both >> in terms of individual people and/or individual events, >> there must be a better way to quickly identify and >> isolate these people and events other than putting the >> entire world on lockdown. >> >> I would imagine that some of the work on fractals should >> be useful to look at this superspreader phenomenon. >> >> Perhaps fractures in glass or metal may have relevance, >> as it may only take a single flaw in a crystal structure >> to destroy the entire structure.

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