SSGAXY versus SSLLA and SSA (shortest splitline variants)
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Warren D Smith
Jun 28, 2017, 3:10:55 PM6/28/17
to electio...@googlegroups.com
SSLLA enjoys these four real world advantages which would not apply to
an abstract spherical world:
1. In the real world nobody cares about the regions near the two poles
which cause difficulty for SSLLA's "two orthogonal directions" view.
2. In the real world a lot of states already have borders defined in part by
using parallels or meridians.
3. Real humans seem to have some sort of inbuilt love for partitioning
areas using cutlines parallel to the x and y axes only. This seems clear
if you look at almost all architecture from the days of the ancient
Babylonians until now.
(This human psychological trait for some reason is not mirrored by
wasps, bees, naked mole rats, birds building nests, and other
home-building animals, incidentally, probably
related to their construction techniques differing from human techniques.)
4. Humans can for many purposes regard the world as locally flat,
without getting hurt much by making that wrong assumption.
So: if such an XY partitioning is done, real humans may feel some sort of
inbuilt love for it, feeling that somehow it
cannot be the work of an evil gerrymanderer.
With SSLLA there is less freedom than allowing arbitrarily-oriented geodesics
like in SSA, and that lesser freedom means SSLLA optimizes length more poorly,
at least locally. And parallels are NOT geodesics, they are curved off
them, which is
clearly suboptimal at least locally. But the flip side of these same facts is
the lesser freedom means SSLLA can run faster; and perhaps globally if
not locally,
things often will seem nicer. A reason to think that is this theorem:
with uniform
population density, and after enough recursive SSLLA splitting, nearly
all districts
will look like AxB rectangles with side-ratio at most sqrt(2)=1.41...
meaning nearly every
district's shape will have "bounded badness."
Meanwhile same assumptions using SSA, it is more puzzling what will
happen although
I think bounded badness still would happen. If SSA happens to start
with a rectangular
state filled with uniform population density, then it will actually do
exactly the
same thing as SSLLA forever,
provided we are on a flat earth and state initially had axis-parallel
orientation.
SSA will however behave better than SSLLA in that scenario if
the initial rectangle orientation was tilted.
And the fact that parallels are geodesically curved is
because of (4) not so horrible provided all states small enough
so that nobody travels along them for
too large a distance.
The pictures we'd made of SSA districtings were using "gnomonic projection"
which projects all geodesics on the sphere to straight lines on the plane map,
and vice versa.
That suggests the following
SSGAXY algorithm (Shortest splitline gnomonic XY):
1. start with initial outline of state.
2. map it to plane using gnomonic projection centered at the
center of the state. (Question: What should the definition of "center" be
for this purpose?)
3. set up an XY coordinate system in this plane. (Question: what orientation?)
4. Now proceed with recursive splittings like the SSA,
but always cutting along an axis-parallel line,
and always using the shortest splitline, where "shortest" assessed
using distance measured on
the sphere using geodesic distance (and note: lines in the plane
always correspond to geodesics on the sphere).
Note, I have left 2 questions unanswered, which need to be answered
to get a concrete realization. E.g. one possibility is to choose the answers
which at the end of the process yield the shortest total length of
all cuts used in the map. That is some sense is the best possible answer-pair,
but would require an outer 3-dimensional
optimization, which might be slow.
A cheaper and simpler answer would be
to make the "center" be the mean location of all human residents,
or the center of the smallest circle on the earth surface that
encloses them all;
and to make the XY orientation be so that variance in X-coordinate among all
humans on the flat map, is maximal.
SSGAXY tries to steal the good virtues of SSLLA while
suffering fewer criticisms. Namely SSGAXY cuts are always
following geodesics on the sphere, so districts are
always convex geodesic-arc-sided polygons. (Unlike SSLLA where
parallels are not geodesics and we get nonconvexity.)
We still enjoy whatever human inbuilt love there is for XY.
We still enjoy higher speed since only 2 cut directions permitted.
SSGAXY runs into trouble if initial country is too large to be
contained in any single hemisphere, because the
gnomonic projection can only handle a hemisphere.
However, on the actual real world this issue does not arise.
Indeed, I have verified that the whole of the (connected landmass)
asia+europe+africa does fit within a single hemisphere.
--
Warren D. Smith
http://RangeVoting.org <-- add your endorsement (by clicking
"endorse" as 1st step)
an abstract spherical world:
1. In the real world nobody cares about the regions near the two poles
which cause difficulty for SSLLA's "two orthogonal directions" view.
2. In the real world a lot of states already have borders defined in part by
using parallels or meridians.
3. Real humans seem to have some sort of inbuilt love for partitioning
areas using cutlines parallel to the x and y axes only. This seems clear
if you look at almost all architecture from the days of the ancient
Babylonians until now.
(This human psychological trait for some reason is not mirrored by
wasps, bees, naked mole rats, birds building nests, and other
home-building animals, incidentally, probably
related to their construction techniques differing from human techniques.)
4. Humans can for many purposes regard the world as locally flat,
without getting hurt much by making that wrong assumption.
So: if such an XY partitioning is done, real humans may feel some sort of
inbuilt love for it, feeling that somehow it
cannot be the work of an evil gerrymanderer.
With SSLLA there is less freedom than allowing arbitrarily-oriented geodesics
like in SSA, and that lesser freedom means SSLLA optimizes length more poorly,
at least locally. And parallels are NOT geodesics, they are curved off
them, which is
clearly suboptimal at least locally. But the flip side of these same facts is
the lesser freedom means SSLLA can run faster; and perhaps globally if
not locally,
things often will seem nicer. A reason to think that is this theorem:
with uniform
population density, and after enough recursive SSLLA splitting, nearly
all districts
will look like AxB rectangles with side-ratio at most sqrt(2)=1.41...
meaning nearly every
district's shape will have "bounded badness."
Meanwhile same assumptions using SSA, it is more puzzling what will
happen although
I think bounded badness still would happen. If SSA happens to start
with a rectangular
state filled with uniform population density, then it will actually do
exactly the
same thing as SSLLA forever,
provided we are on a flat earth and state initially had axis-parallel
orientation.
SSA will however behave better than SSLLA in that scenario if
the initial rectangle orientation was tilted.
And the fact that parallels are geodesically curved is
because of (4) not so horrible provided all states small enough
so that nobody travels along them for
too large a distance.
The pictures we'd made of SSA districtings were using "gnomonic projection"
which projects all geodesics on the sphere to straight lines on the plane map,
and vice versa.
That suggests the following
SSGAXY algorithm (Shortest splitline gnomonic XY):
1. start with initial outline of state.
2. map it to plane using gnomonic projection centered at the
center of the state. (Question: What should the definition of "center" be
for this purpose?)
3. set up an XY coordinate system in this plane. (Question: what orientation?)
4. Now proceed with recursive splittings like the SSA,
but always cutting along an axis-parallel line,
and always using the shortest splitline, where "shortest" assessed
using distance measured on
the sphere using geodesic distance (and note: lines in the plane
always correspond to geodesics on the sphere).
Note, I have left 2 questions unanswered, which need to be answered
to get a concrete realization. E.g. one possibility is to choose the answers
which at the end of the process yield the shortest total length of
all cuts used in the map. That is some sense is the best possible answer-pair,
but would require an outer 3-dimensional
optimization, which might be slow.
A cheaper and simpler answer would be
to make the "center" be the mean location of all human residents,
or the center of the smallest circle on the earth surface that
encloses them all;
and to make the XY orientation be so that variance in X-coordinate among all
humans on the flat map, is maximal.
SSGAXY tries to steal the good virtues of SSLLA while
suffering fewer criticisms. Namely SSGAXY cuts are always
following geodesics on the sphere, so districts are
always convex geodesic-arc-sided polygons. (Unlike SSLLA where
parallels are not geodesics and we get nonconvexity.)
We still enjoy whatever human inbuilt love there is for XY.
We still enjoy higher speed since only 2 cut directions permitted.
SSGAXY runs into trouble if initial country is too large to be
contained in any single hemisphere, because the
gnomonic projection can only handle a hemisphere.
However, on the actual real world this issue does not arise.
Indeed, I have verified that the whole of the (connected landmass)
asia+europe+africa does fit within a single hemisphere.
--
Warren D. Smith
http://RangeVoting.org <-- add your endorsement (by clicking
"endorse" as 1st step)
Warren D Smith
Jun 28, 2017, 4:03:51 PM6/28/17
to electio...@googlegroups.com
> I have verified that the whole of the (connected landmass)
> asia+europe+africa does fit within a single hemisphere.
--the center of that hemisphere needs to be located about
> asia+europe+africa does fit within a single hemisphere.
the location of Kuwait, then the whole landmass fits in a circle of
angular radius about 80 degrees.
Thus verifying what we always suspected, Kuwait is the
center of the world.
Brian Langstraat
Jun 28, 2017, 6:49:26 PM6/28/17
to The Center for Election Science
SSGAXY tries to steal the good virtues of SSLLA while
suffering fewer criticisms.
Thief! How dare you try to improve an idea that I had while trying to improve on an idea that you had!
Shortest Splitarc Latitude/Longitude Algorithm (SSLLA) is discussed in the following Topics:
Real humans seem to have some sort of inbuilt love for partitioning
areas using cutlines parallel to the x and y axes only.
It's hip to be square.
(This human psychological trait for some reason is not mirrored by
wasps, bees, naked mole rats, birds building nests, and other
home-building animals, incidentally, probably
related to their construction techniques differing from human techniques.)
Squares (rectangular polygons) are very efficient at covering a surface.
Hexagons are the only more efficient polygon, so bees use them in hives.
If SSLLA or SSGAXY could be modified to created hexagonal districts, we may have the perfect redistricting algorithm.
So: if such an XY partitioning is done, real humans may feel some sort of
inbuilt love for it, feeling that somehow it
cannot be the work of an evil gerrymanderer.
Districts created using SSLLA would look like the ideal districts we picture in our dreams until we awake to the gerrymandering nightmare.
2. map it to plane using gnomonic projection centered at the
center of the state. (Question: What should the definition of "center" be
for this purpose?)
A state's borders rarely change, so the "center of mass" of the state's elliptical surface would work well.
(Unlike SSLLA where
parallels are not geodesics and we get nonconvexity.)
The general public and courts may find districts created using SSGAXY to be confusing, since they would have slight curves when compared to political boundaries created using lat/long lines.
Warren D Smith
Jun 28, 2017, 7:59:57 PM6/28/17
to electio...@googlegroups.com
> The general public and courts may find districts created using SSGAXY to be
> confusing, since they would have slight curves when compared to political
> boundaries created using lat/long lines.
--if the maps are drawn using gnomonic projection, SSGAXY
> confusing, since they would have slight curves when compared to political
> boundaries created using lat/long lines.
cuts will involve only straight lines!
SSLLA will involve only straight lines too, provided its maps
are drawn using the mercator (conformal) projection, or
the orthogonal-cylindrical (equi-areal) projection, both using pole-pole axis.
In each class of projections, the other class of cuts will involve curved
lines.
Brian Langstraat
Jun 29, 2017, 10:59:04 AM6/29/17
to The Center for Election Science
> The general public and courts may find districts created using SSGAXY to be
> confusing, since they would have slight curves when compared to political
> boundaries created using lat/long lines.
--if the maps are drawn using gnomonic projection, SSGAXY
cuts will involve only straight lines!
To clarify, the northern and southern edges of the districts (excluding state boundaries) created using the Shortest Splitline Gnomonic Algorithm XY (SSGAXY) would not be parallel to the latitude lines on the earth.
The general public and some courts may not understand why these edges deviate from other political boundaries created using lat/long lines.
By definition, the northern and southern edges of the districts (excluding state boundaries) created using the Shortest Splitarc Latitude/Longitude Algorithm (SSLLA) would be parallel to the latitude lines on the earth.
Since humans (who seem to like rectangles created by grid lines) live on the Earth (approximated by an ellipsoid with standardized latitude and longitude lines), I propose that the SSLLA is the best redistricting algorithm to use on Earth.
I propose that either the SSGAXY or the Ellipsoid Shortest Splitline Algorithm (ESSA) is the best redistricting algorithm to use for all of the remaining celestial bodies in the universe.
Deal?
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