[math-fun] measuring the irregularity (or salamander-ness) of a planar region
Is there a good measure of how irregular a region of the plane is? something that gives 0, say, for a circular disk, and 1 (or infinity) for a fractal gasket? the motivation is to find a measure of how irregular (or gerrymandered) a congressional district is. Neil -- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
a nice one for Steven Finch? He did work on the 'HullMeanWidth' for convex polytopes in 3D. This has a similar flavour: circumference (alias 'outer boundary length') / circumference of equal-area circle ? Wouter. -----Original Message----- From: Neil Sloane Sent: Friday, June 20, 2014 11:38 PM To: fun Subject: [math-fun] measuring the irregularity (or salamander-ness) of aplanar region Is there a good measure of how irregular a region of the plane is? something that gives 0, say, for a circular disk, and 1 (or infinity) for a fractal gasket? the motivation is to find a measure of how irregular (or gerrymandered) a congressional district is. Neil -- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Area of convex hull divided by area of region? On Fri, Jun 20, 2014 at 6:21 PM, Wouter Meeussen <wouter.meeussen@telenet.be
wrote:
a nice one for Steven Finch? He did work on the 'HullMeanWidth' for convex polytopes in 3D. This has a similar flavour:
circumference (alias 'outer boundary length') / circumference of equal-area circle ?
Wouter.
-----Original Message----- From: Neil Sloane Sent: Friday, June 20, 2014 11:38 PM To: fun Subject: [math-fun] measuring the irregularity (or salamander-ness) of aplanar region
Is there a good measure of how irregular a region of the plane is?
something that gives 0, say, for a circular disk, and 1 (or infinity) for a fractal gasket?
the motivation is to find a measure of how irregular (or gerrymandered) a congressional district is.
Neil
-- Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Use several different lengths of "ruler" to measure the circumference and from that obtain an average fractal dimension. However if such districts are anything like in the UK then irregularity is more down to local geography and population density/location so I don't see how it could be used to detect politically motivated adjustment of boundaries - unless cross-referenced with even more local data relating to which locatuons voted for what. On 20 Jun 2014, at 23:27, Allan Wechsler wrote:
Area of convex hull divided by area of region?
On Fri, Jun 20, 2014 at 6:21 PM, Wouter Meeussen <wouter.meeussen@telenet.be
wrote:
a nice one for Steven Finch? He did work on the 'HullMeanWidth' for convex polytopes in 3D. This has a similar flavour:
circumference (alias 'outer boundary length') / circumference of equal-area circle ?
Wouter.
-----Original Message----- From: Neil Sloane Sent: Friday, June 20, 2014 11:38 PM To: fun Subject: [math-fun] measuring the irregularity (or salamander-ness) of aplanar region
Is there a good measure of how irregular a region of the plane is?
something that gives 0, say, for a circular disk, and 1 (or infinity) for a fractal gasket?
the motivation is to find a measure of how irregular (or gerrymandered) a congressional district is.
Neil
-- Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.
Diameter divided by radius? Circumference divided by area? Charles Greathouse Analyst/Programmer Case Western Reserve University On Fri, Jun 20, 2014 at 5:38 PM, Neil Sloane <njasloane@gmail.com> wrote:
Is there a good measure of how irregular a region of the plane is?
something that gives 0, say, for a circular disk, and 1 (or infinity) for a fractal gasket?
the motivation is to find a measure of how irregular (or gerrymandered) a congressional district is.
Neil
-- Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
For a smooth simply connected region of area A = 1, I would use something like the standard deviation s of the curvature of the boundary. This would be 0 only for a circle. For areas A <> 1, the figure should be s/sqrt(A) to scale right. For piecewise-smooth boundary curves, the figure should be the limit of smooth curves approximating the boundary in the appropriate sense. For more general boundary curves, it should be the limit of appropriately approximating piecewise-smooth curves. (This could all be expressed with no reference to smoothness, but I omit the details.) --Dan Neil wrote: ----- Is there a good measure of how irregular a region of the plane is? something that gives 0, say, for a circular disk, and 1 (or infinity) for a fractal gasket? the motivation is to find a measure of how irregular (or gerrymandered) a congressional district is. -----
="Neil Sloane" <njasloane@gmail.com> Is there a good measure of how irregular a region of the plane is?
This is an interesting pure math question, which I'll leave t' y'all.
the motivation is to find a measure of how irregular (or gerrymandered) a congressional district is.
This application raises a different and wider topic that begs for better definition. Hopefully my analysis of the underlying problem definition will be sufficiently mathy and fun. If not please ignore with my apologies. Beyond calculating the salamander-nesses of various candidate districtings, we also must establish what the optimization criteria are by which to rank them. To address that we need to ask: just what problem is "boundary delimitation" trying to solve? Traditionally the process appears to involve two distinct operations: 1. Assign each individual to a set, call this their "electoral district" 2. Assign each point in a region to a set, call this its "geo-district" The result of #1 is the legislatively significant one, since electoral districts are allocated representatives to vote and otherwise act on behalf of the entire set. In contrast, #2 seems to be more of a (seemingly antiquated?) hack by which to implement #1, namely by associating geo-districts to electoral districts. The consequences and costs of this hack should be re-evaluated. ======== Take "fairness", for instance: One optimization criteria appears to be that folks want the result of #1 to be "fair". This seems to be interpreted to mean that the distribution of individuals within electoral districts matches that of the whole population (within random statistical sampling variance). In order to implement a "fairness measure" for use as an optimization goal, current practice seems to be to choose some arbitrary set of individual features and measure their uniformity. A perennial source of contention is choosing which features (such as party affiliation, race, language etc) are deemed relevant, and which features are suppressed (such as religion, IQ, favorite sports or drug allergies). This is inevitably a highly political question. There's a bug with representing individuals by groups, because as more and more features are added to ensure uniform representation the subpopulations defined by any given feature vector get smaller and smaller until finally they lose statistical meaning, since every individual is by definition unique once enough dimensions are used to classify them. Reductio ad absurdum (or maybe not) one could argue that, say, a-priori insurance rate settings based on a panel of classifiers are unfairly discriminatory because there is zero historical basis to which any actuarial science can be applied. Hence, because geographical distribution of traits is naturally non-uniform (eg language groups tend to affiliate in neighborhoods because it makes everyday intercourse easier), with enough identity-political balkanization in feature space any geo-districting scheme can be charged as "unfair". -------- So let's ask: why not just RANDOMLY assign people to political districts? By construction this should result in the FAIREST possible uniformity of representation. Geo-districting seems as quaintly dated as gluing colored stamps on paper envelopes and leaving them in boxes outside for liveried postal elves to whisk away. You could use a lottery, or SSN's mod X, or whatever (Kurt Vonnegut amusingly suggested giving everyone an arbitrary middle name, such as "flower" or "whale", so no one would be lonely for want of affiliation). On Election Day everyone in your group, call it District 9 say, would vote, and whoever wins would be District 9's electoral representative. No special interest advocacy, no endless arguing over gerrymandering minutiae, no hard math puzzles about smoothing out geographical subdivisions would be needed. And that would be "fair", right? ======== The problem of course is that random assignment merely results in a "quantized" equivalent of pure democracy, with the same attendant "tyranny of the majority" and other well-known horrors of inequity. People WANT non-uniform concentrations of representation, so that they can form more effective advocacy groups around their pet issues of vital importance to them. Which groups get to exist and prevail over others on irreconcilable differences is the essence of politics. Historically there was more of a tendency for like-minded subgroups to geographically concentrate, either by "right" of prior conquest or from people emigrating away from locations they found less congenial. Thus geo-districting was adopted, and has persisted. It is unsurprising that any such hoary institution will be gamed and gamed again as long as there are interested parties constantly pushing and pulling its details into fractal pretzels by means of incremental legislation. However simply low-pass filtering gerrymandered boundaries doesn't address the complete design problem, which is finding the "dialectic synthesis" between the desire for general fairness versus the necessary inevitability of irreconcilable interests. 'nuff said, I'll leave that puzzle t' y'all too.
I've imagined the following scheme. Voters select their favorite candidates; all candidates are at-large. Voters vote *one at a time*. As soon as a candidate receives N votes, they are elected and their name is removed from consideration, so all elected candidates get exactly N votes. On Sat, Jun 21, 2014 at 2:23 PM, Marc LeBrun <mlb@well.com> wrote:
="Neil Sloane" <njasloane@gmail.com> Is there a good measure of how irregular a region of the plane is?
This is an interesting pure math question, which I'll leave t' y'all.
the motivation is to find a measure of how irregular (or gerrymandered) a congressional district is.
This application raises a different and wider topic that begs for better definition.
Hopefully my analysis of the underlying problem definition will be sufficiently mathy and fun. If not please ignore with my apologies.
Beyond calculating the salamander-nesses of various candidate districtings, we also must establish what the optimization criteria are by which to rank them.
To address that we need to ask: just what problem is "boundary delimitation" trying to solve?
Traditionally the process appears to involve two distinct operations: 1. Assign each individual to a set, call this their "electoral district" 2. Assign each point in a region to a set, call this its "geo-district"
The result of #1 is the legislatively significant one, since electoral districts are allocated representatives to vote and otherwise act on behalf of the entire set.
In contrast, #2 seems to be more of a (seemingly antiquated?) hack by which to implement #1, namely by associating geo-districts to electoral districts.
The consequences and costs of this hack should be re-evaluated.
======== Take "fairness", for instance:
One optimization criteria appears to be that folks want the result of #1 to be "fair". This seems to be interpreted to mean that the distribution of individuals within electoral districts matches that of the whole population (within random statistical sampling variance).
In order to implement a "fairness measure" for use as an optimization goal, current practice seems to be to choose some arbitrary set of individual features and measure their uniformity.
A perennial source of contention is choosing which features (such as party affiliation, race, language etc) are deemed relevant, and which features are suppressed (such as religion, IQ, favorite sports or drug allergies). This is inevitably a highly political question.
There's a bug with representing individuals by groups, because as more and more features are added to ensure uniform representation the subpopulations defined by any given feature vector get smaller and smaller until finally they lose statistical meaning, since every individual is by definition unique once enough dimensions are used to classify them.
Reductio ad absurdum (or maybe not) one could argue that, say, a-priori insurance rate settings based on a panel of classifiers are unfairly discriminatory because there is zero historical basis to which any actuarial science can be applied.
Hence, because geographical distribution of traits is naturally non-uniform (eg language groups tend to affiliate in neighborhoods because it makes everyday intercourse easier), with enough identity-political balkanization in feature space any geo-districting scheme can be charged as "unfair".
--------
So let's ask: why not just RANDOMLY assign people to political districts? By construction this should result in the FAIREST possible uniformity of representation.
Geo-districting seems as quaintly dated as gluing colored stamps on paper envelopes and leaving them in boxes outside for liveried postal elves to whisk away.
You could use a lottery, or SSN's mod X, or whatever (Kurt Vonnegut amusingly suggested giving everyone an arbitrary middle name, such as "flower" or "whale", so no one would be lonely for want of affiliation).
On Election Day everyone in your group, call it District 9 say, would vote, and whoever wins would be District 9's electoral representative. No special interest advocacy, no endless arguing over gerrymandering minutiae, no hard math puzzles about smoothing out geographical subdivisions would be needed.
And that would be "fair", right? ========
The problem of course is that random assignment merely results in a "quantized" equivalent of pure democracy, with the same attendant "tyranny of the majority" and other well-known horrors of inequity.
People WANT non-uniform concentrations of representation, so that they can form more effective advocacy groups around their pet issues of vital importance to them. Which groups get to exist and prevail over others on irreconcilable differences is the essence of politics.
Historically there was more of a tendency for like-minded subgroups to geographically concentrate, either by "right" of prior conquest or from people emigrating away from locations they found less congenial.
Thus geo-districting was adopted, and has persisted. It is unsurprising that any such hoary institution will be gamed and gamed again as long as there are interested parties constantly pushing and pulling its details into fractal pretzels by means of incremental legislation.
However simply low-pass filtering gerrymandered boundaries doesn't address the complete design problem, which is finding the "dialectic synthesis" between the desire for general fairness versus the necessary inevitability of irreconcilable interests.
'nuff said, I'll leave that puzzle t' y'all too.
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On 6/21/2014 11:23 AM, Marc LeBrun wrote:
="Neil Sloane" <njasloane@gmail.com> Is there a good measure of how irregular a region of the plane is? This is an interesting pure math question, which I'll leave t' y'all.
the motivation is to find a measure of how irregular (or gerrymandered) a congressional district is. This application raises a different and wider topic that begs for better definition.
Hopefully my analysis of the underlying problem definition will be sufficiently mathy and fun. If not please ignore with my apologies.
Beyond calculating the salamander-nesses of various candidate districtings, we also must establish what the optimization criteria are by which to rank them.
To address that we need to ask: just what problem is "boundary delimitation" trying to solve?
Traditionally the process appears to involve two distinct operations: 1. Assign each individual to a set, call this their "electoral district" 2. Assign each point in a region to a set, call this its "geo-district"
The result of #1 is the legislatively significant one, since electoral districts are allocated representatives to vote and otherwise act on behalf of the entire set.
In contrast, #2 seems to be more of a (seemingly antiquated?) hack by which to implement #1, namely by associating geo-districts to electoral districts.
The consequences and costs of this hack should be re-evaluated.
======== Take "fairness", for instance:
One optimization criteria appears to be that folks want the result of #1 to be "fair". This seems to be interpreted to mean that the distribution of individuals within electoral districts matches that of the whole population (within random statistical sampling variance).
What "whole population" - the nations, the states, the...
In order to implement a "fairness measure" for use as an optimization goal, current practice seems to be to choose some arbitrary set of individual features and measure their uniformity.
A perennial source of contention is choosing which features (such as party affiliation, race, language etc) are deemed relevant, and which features are suppressed (such as religion, IQ, favorite sports or drug allergies). This is inevitably a highly political question.
There's a bug with representing individuals by groups, because as more and more features are added to ensure uniform representation the subpopulations defined by any given feature vector get smaller and smaller until finally they lose statistical meaning, since every individual is by definition unique once enough dimensions are used to classify them.
Reductio ad absurdum (or maybe not) one could argue that, say, a-priori insurance rate settings based on a panel of classifiers are unfairly discriminatory because there is zero historical basis to which any actuarial science can be applied.
Hence, because geographical distribution of traits is naturally non-uniform (eg language groups tend to affiliate in neighborhoods because it makes everyday intercourse easier), with enough identity-political balkanization in feature space any geo-districting scheme can be charged as "unfair".
--------
So let's ask: why not just RANDOMLY assign people to political districts? By construction this should result in the FAIREST possible uniformity of representation.
I'm not sure what sense of 'fair' is implemented here. Why not let people chose what district they vote in, or what has the same effect, elect all the representatives at-large using something like rank voting.
Geo-districting seems as quaintly dated as gluing colored stamps on paper envelopes and leaving them in boxes outside for liveried postal elves to whisk away.
You could use a lottery, or SSN's mod X, or whatever (Kurt Vonnegut amusingly suggested giving everyone an arbitrary middle name, such as "flower" or "whale", so no one would be lonely for want of affiliation).
On Election Day everyone in your group, call it District 9 say, would vote, and whoever wins would be District 9's electoral representative. No special interest advocacy, no endless arguing over gerrymandering minutiae, no hard math puzzles about smoothing out geographical subdivisions would be needed.
And that would be "fair", right?
Assuming large numbers it would have the effect that every district would be dominated by the majority, so there would be no black, gay, or atheist representatives. This was exactly the effect achieved by gerrymandering in the south before the civil rights act.
========
The problem of course is that random assignment merely results in a "quantized" equivalent of pure democracy, with the same attendant "tyranny of the majority" and other well-known horrors of inequity.
People WANT non-uniform concentrations of representation, so that they can form more effective advocacy groups around their pet issues of vital importance to them. Which groups get to exist and prevail over others on irreconcilable differences is the essence of politics.
Historically there was more of a tendency for like-minded subgroups to geographically concentrate, either by "right" of prior conquest or from people emigrating away from locations they found less congenial.
Thus geo-districting was adopted, and has persisted. It is unsurprising that any such hoary institution will be gamed and gamed again as long as there are interested parties constantly pushing and pulling its details into fractal pretzels by means of incremental legislation.
However simply low-pass filtering gerrymandered boundaries doesn't address the complete design problem, which is finding the "dialectic synthesis" between the desire for general fairness versus the necessary inevitability of irreconcilable interests.
I agree. Brent Meeker
'nuff said, I'll leave that puzzle t' y'all too.
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participants (8)
-
Allan Wechsler -
Charles Greathouse -
Dan Asimov -
David Makin -
Marc LeBrun -
meekerdb -
Neil Sloane -
Wouter Meeussen