Your Dumbest Theory

[mfbrandi]

2 hours ago, scott-martin said:

Well, on the one hand:

eyes closed;

eyes open;

eyes thrown in the air and juggled.

But on the other, these look like the glyphs used to notate Gloranthan choreography. “Young elf, there’s no need to feel down … It’s fun to stray in the True Illusory Forest.” Or maybe they are pom-poms and that is something by Toni Basil. Pom-poms or no, “Crosseyed and Painless” might soundtrack a Gloranthan rebirth.

Spoiler

Lost my shape, trying to act casual
Can’t stop, I might end up in the hospital
Changing my shape, I feel like an accident
They’re back to explain their experience

Isn’t it weird? Looks too obscure to me
Wasting away, that was their policy

I’m ready to leave, I push the facts in front of me
Facts lost, facts are never what they seem to be
Nothing there, no information left of any kind
Lifting my head, looking for the danger signs

There was a line, there was a formula
Sharp as a knife, facts cut a hole in us

2 hours ago, scott-martin said:

You know what they say about Oedipus and eye gods . . . Arkat goes out of his way to join (or construct) ZZ.

Well, every boy crazy ’bout a self-made man, no? I feel we are drifting deeper into Gloranthan karaoke: ZeeZee is doing his warm-up exercises.

Spoiler

Harvester of eyes, that's me
And I see all there is to see
When I look inside your head
Right up front to the back of your skull
Well, that's my sign that you are dead
My list for you checks off as null
I'm the harvester of eyes!

I'm the eye man of TV
With my ocular TB

I need all the peepers I can find
Inside the barn where you find the hay
Just last week, I took a ride
So high on eyes, I almost lost my way
I'm the harvester of eyes!

2 hours ago, scott-martin said:

(secret figure)

The secret figure? The rune no one is talking about — there being a notable absence of true binary thinking? Negation: ¬

This yields ¬

Some say this means unlawful killing, because it’s murder on the dance floor.

Spoiler

    It’s murder on the dance floor
    You’d better not steal the moves
    DJ, gonna burn this goddamn house right down

Sophie Ellis Bextor is a secret Arkati warning us that the true Arkati pulls up the ladder behind her — no heroquesting and illumination secrets for hoi polloi. Gully Foyle begs to differ, and he has Trickster on his side. The cat’s long out of the bag.

But old Bill tells me that scientists have always said there is no such thing as a soul; now they are in a position to prove it. Seems a more usable insight.

[scott-martin]

Dance dance dance till you're dead . . . 

Since we all know  is also read "torch" in some forests then the move notated as illusion over truth is also known as "toasting the seeds" or "Jack Flash (sat on a candlestick)" and is what drives a surprising number of modern ethnobotanists into the field.

The null is a good one. If I were analyzing young Arkat right here I'd say his fundamental problem when confronted with the rock of law and the sword of action is that he read a null into the dynamic and went looking for it until he found it, dooming the old world in the process. The rock and the sword aren't opposed exactly. They're just aspects of the same complex phenomenon. But under the influence of those jack flash elves his mind leapt to the cosmic crisis and here we are.
 

On 6/5/2023 at 12:56 PM, mfbrandi said:

old Bill

Scene from the Monster Empire:



The problem with performing terminal lunar experience at the table is that the sphere of personal agency expressed by the character sheet is both much smaller than the conventional game and much larger when you map out all their attachments and lozengoidal concerns. Jack Flash is here!

[JRE]

Thanks for the Burroughs McNeill link. Suddenly many things about Glorantha make more sense.

 

This could be Dorastor, or the monster Empire, but I will just go with the Godtime itself.

Edited June 7, 2023 by JRE
Recognizing McNeill

[mfbrandi]

2 hours ago, JRE said:

This could be Dorastor, or the monster Empire, but I will just go with the Godtime itself.

Quite right: take ’em at their word — no sequence in Godtime, so any sample is chock full of the worst of the Gods War (and everything else, admittedly). There is no safe haven. It is the nightmare interpretation of Schrödinger’s Cat Carrier — a box whose insides don’t bear thinking about.

Time’s a doddle in comparison. Sure we’re slowly sliding into the void (nothing to be dreaded), but there are plenty of good times left. It is a shame that the idiots keep trying to blow it up in attempts to Make Glorantha Great Again.

Do we say time is opening the box, finding a live cat, and then coming to terms with the fact that the cat will die eventually?

Hmm … “chock full of the worst of the Gods War and everything else”. Is that the secret of Dorastor, that it is a little bubble of Godtime? Ralzakark a palimpsest of the best and the worst a demigod can be? Leakage from other planes has turned Dorastor into one of the Strugatskis’ Zones?

[mfbrandi]

3 hours ago, scott-martin said:

The rock and the sword aren't opposed exactly.

No, not exactly: they are points on a triangle. If = rock and = scissors, then what is paper?

(as we have noted before) is a crutch. The eternal is a crutch for those who cannot accommodate to the temporary, the provisional, and the contingent. Those reaching for the noumenal tend to treat the phenomenal world quite shabbily. “Their eyes were on God” — oh, dear! 😉

[scott-martin]

26 minutes ago, mfbrandi said:

paper

[mfbrandi]

2 minutes ago, scott-martin said:

Paper is undead trees sealed with kaolin?

No longer will anyone order the dead tree edition — from now on, the zombie tree edition.

[mfbrandi]

 This got me thinking about runes and symmetry:

On 6/5/2023 at 7:01 PM, M Helsdon said:

Peloria as a term in biology derives from … the Greek word pelōros, meaning 'monstrous'.

Quote

Peloria or a peloric flower is the aberration in which a plant that normally produces zygomorphic flowers produces actinomorphic flowers instead. — Floral Symmetry

So if we generalise and have increasing orders of rotational symmetry — especially if produced from something with lower order symmetry — be the indicator of monstrousness, the circular runes would seem to be most easily associated with monstrousness.

has only bilateral symmetry, not rotational. Like , , and (there are more).

Some runes have order 2 rotational symmetry — i.e. only fit into themselves at 0°/360° and 180° — and so (on this bizarre line of thinking) wouldn’t be monstrous: , , , ,

has no rotational symmetry and lacks even bilateral (mirror) symmetry. Appropriately, this puts it at the opposite end of our spectrum from . lacks rotational (and bilateral) symmetry, which for a wheel is mad.

Law I suppose could be derived from the old Force rune (or Dara Happan “sh”?) — something like /\ — which has bilateral symmetry. Rotate smoothly and you get , which some may find appropriate.

, , and arguably (i.e. the ‘normalised’ way I draw it, else it is bilateral like Death and Chaos) are monstrous at order 3. outdoes them at order 4. , , , , and can all be rotated freely and fit into themselves.

Note that by rotating smoothly about the centre of one ‘ball’, you can generate from — the sun: a more monstrous illusion.

I haven’t tried to fit all the runes into this scheme, that is left as an exercise for the student. If you buy into the spectrum of chiral —> bilateral symmetry —> increasing orders of rotational symmetry (with monstrosity kicking in at 3) — and there is really no reason you have to — what does it all mean? If anything!

If Krarsht wants a glyph, rotating works well:

[Joerg]

Storm gains symmetry through movement, absolute point symmetry over Time.

The birth of Umath necessitated the birth of Time?

[mfbrandi]

4 hours ago, Joerg said:

Storm gains symmetry through movement, absolute point symmetry over Time.
The birth of Umath necessitated the birth of Time?

By “point symmetry”, you mean rotational symmetry of order 2, right? (Apologies if not.) But surely you can take two copies of any rune, rotate one, and generate a new composite figure with that symmetry, no?

But must there be pairs of Umath time-slices with this property? One can imagine those “wild and crazy” storm types positively revelling in refusing every kind of symmetry and quasi-symmetry offered. Why not? Unpredictability is their thing. Leave all the cryptic symmetry stuff to the Arkati conspiracy theorists and their well-thumbed Dan Brown paperbacks.

(The wheel of change lacking rotational symmetry still bugs me, though.)

The storm propagandists can then say that it is the extreme rotational symmetry of the sun that is the mark of a monster. The more thoughtful of the storm theorists can say that you create monsters from the bilateral symmetry of chaos by giving it a spin — although everyone will stop saying it when the Storm Bulls start burning crosses on their lawns.

Meanwhile the solar scribes will poo-poo etymologies from languages that have clearly fallen through a wormhole from “another world” and hymn the perfection of symmetry — the more the better — and claim that Umatum is worse than chaos — lacking even its primitive bilateral symmetry — and worse than Krarsht, who at least had the decency to adapt to Yelm’s cosmos by giving her void a twist to produce order 3 rotational symmetry. Nothing on the perfection of a circle, obviously, but at least she showed willing. Storm is less than nothing.

It is nice that moon and chaos sit between the extremes of sun and storm. We don’t have to take runic symmetry seriously, just wind up the Gloranthan scribes and let them throw dung at each other.

[mfbrandi]

Is it enlightenment and Chaos magic that built the Empire, or is it good old-fashioned imperialism that allowed such experiments and cultural flourishing?

In the real world, think of the Athenian Empire enabling philosophy and theatre or — at empire’s sticky end? — the Tories’ “you’ve never had it so good” society setting us up for the Swinging Sixties. It may be ugly, but isn’t that often how it goes? Subsequent feedback loops (real or imagined) may then lead us to flip the direction of causation.

Of course, in fiction, we can simply stipulate the direction of causation. Is this dishonest? Not necessarily, but be careful what you read back into the “real world”.

(This prompted by today’s radio discussion of Oedipus the King and stuff over at Monster Empire.)

[Joerg]

1 hour ago, mfbrandi said:

By “point symmetry”, you mean rotational symmetry of order 2, right? (Apologies if not.)

No. The cyclone is a rotating system, and over a set of full rotations the spiral will have touched everywhere on a radius from the eye with the same coverage if you accumulate the presence.

Orlanth's Ring (a spiral on a spiralling path) might not quite be as balanced.

The antics of the World Storm when the Doldrums suddenly restart in the center after having wandered off far into the West may have been caused by the Sunstop - the missing distance might be used to calculate how much the years were shortened by that transgression.

[scott-martin]

13 hours ago, mfbrandi said:

I think the version that appears now in standard Holy Country runic fonts was deliberately altered ("changed") from the more uniform triskelion as part of a larger development of the old Larnste mysteries to favor modern (or future) Mastakos at the Traveling Stone. But this is something that will emerge in play, which is why the wheel currently only spins with effort unless we assume an implied higher dimension or blur like in a Wyndham Lewis. Joerg, I love the note on Storm prefiguring time, how have you been?

--

A quick comparison of the demographic data we have for 1616 (orange box) and 1621 (guide) reveals that either no new census was taken or populations from Black Horse County to Kralorela had largely plateaued. This seems to be peak net fertility, the level at which the landscape and modern best magical practices can support a population in dynamic equilibrium: a baby is born, someone older moves on to surrender the space. 

Two things are immediately interesting here. First, most of the elder race enclaves I surveyed are not showing signs of imminent decline except for Dagori Inkarth, which seems to be undergoing some kind of trollkin collapse either related to the way the numbers were calculated or something more fundamental happening in the caves. Ducks in Esrolia also seem to be under pressure. Otherwise, the conventional "elves are going away to make room for modern " trope is just not happening on the ground yet. If anything, the hsunchen population in the east is increasing but this could be an artifact of some territorial cycle we don't currently understand.

Second, when we start seeing fresh numbers for 1625 and beyond we'll be able to gauge the impact and recovery from the various catastrophes of the early hero wars. If for example the Sartar stats are roughly where they were in the guide era, we'll know the Winter Of Our Discontent really didn't do much in the long term beyond pushing the age curve down a few years (crib death to baby boom). Ideally we'll be able to spot refugee populations. And in "control" regions where nothing much has happened yet (Kralorela) we should get a better view of whether populations across the lozenge really have peaked or are still inching forward.

But I'm starting to think that unless we see some incremental gains in the quiet places, net fertility (love minus death) has gone as far as it can. This is how many people the world can hold before a forced correction. Call it a cosmological constant. And as for the winter, I'm still not convinced it was anything but a kind of dragonewt dream. The weather was real. The social upheaval left lingering trauma. The faces changed. But did it "happen" in the usual sense?

[mfbrandi]

1 hour ago, Joerg said:

The cyclone is a rotating system, and over a set of full rotations the spiral will have touched everywhere on a radius from the eye with the same coverage if you accumulate the presence.

OK, but I am still not sure what you mean by “absolute point symmetry”. Do you mean that the shape described will be a filled-in circle — that the accumulated figure will have the symmetry of Monster Yelm (or Darkness)? I’m no mathematician, and I am not sure how we would prove that. With our current approximation to the Storm rune and 120 copies rotated and centred (i.e. 3° precision), we get:

It doesn’t look like it is tending toward a circle, but appearances can be deceptive. Iconographically — i.e. playing silly buggers — do we want it to? If so, why?

[jajagappa]

6 hours ago, mfbrandi said:

It doesn’t look like it is tending toward a circle, but appearances can be deceptive.

When storms converge... Eurmal is ready to welcome us.

[Jex]

8 hours ago, mfbrandi said:

It doesn’t look like it is tending toward a circle, but appearances can be deceptive. Iconographically — i.e. playing silly buggers — do we want it to? If so, why?

If the question is whether, if you rotate the Air rune 360 degrees around a point, the shape made by collecting all the points it passes through during the rotation has full rotational symmetry (i.e. is circular), that's certainly true.  It's not hard to prove, though I'm not sure how easy it is to phrase the proof in a way that makes sense to someone without much mathematical background.

(I'll give it a try, though but I'll put it behind spoiler tags because it may get a bit long and it's not very interesting.)
 

Spoiler

Okay, this is one of those things that seems intuitively obvious, but isn't necessarily trivial to rigorously prove.  I'm writing this quickly off the top of my head so it's going to get a bit messy and given more time I could probably come up with a more concise explanation, but anyway... Consider any point P on the air rune, and take its distance from the center of rotation O.  Now, consider any other point Q the same distance from the center of rotation.  Draw line segments from each point to the center (PO and QO), and measure the angle A between the lines (clockwise from the first to the second).  Rotating point P by angle A around the center of rotation overlays it on point Q.  (Hopefully this is obvious; if not it can be shown from the mathematical definition of rotation, but I don't want to make this more technical than it already is.)  This means that during the course of its 360 degree rotation, the air rune passes through point Q, which means point Q is in the final shape formed by the rotation.  Since the only criterion for choosing Q was that it was the same distance as P from the center of rotation, this is true for all points the same distance as P from the center of rotation.  By the definition of a circle, this set of point describes a circle centered at O.  (The most common way to define a circle mathematically is as the set of all points the same distance from a given point (the center of the circle).)  So the final shape includes a circle centered at point O and passing through point P.  Since point P was any point on the air rune, the final shape includes circles centered at point O and passing through point P.

Does the shape include any points not on such a circle?  No, because consider any point Q' in the final rotational shape.  The final rotational shape is defined as all the points the air rune passes through as it rotates 360 degrees.  If Q' is in the final rotational shape, then some point P' on the air rune must pass through it at some point during the rotation.  But rotation preserves the distance to the center of rotation, so for some P' to pass through Q' they must be the same distance from the origin... which means, again from the definition of a circle, that Q' lies on a circle centered on the origin passing through P'.

Therefore, the final shape formed by all the points the rune passes through can be considered a collection of circles centered at the center of rotation O.  Since a circle has full rotational symmetry (as well as point symmetry) about its center, a collection of concentric circles has the same symmetry, so the final rotational shape also has such symmetry, Q.E.D.

The problem here is, though, that that's not a property unique to the air rune.  Note that nothing in the proof referred to anything specific to the air rune—this is true for any shape at all!  Rotate any collection of points about a particular center of rotation, and the collection of all points they pass through is going to have circular symmetry.  Nothing about the air rune is special in that regard.

(Not all shapes, however, will of course give solid filled circles... rotate the illusion rune or the law rune about the center and you'll get a hollow circle, but it'll still have full rotational (and point) symmetry.)

Ah, wait, no, hold on; there's more to it than that.  @Joerg specifies that "over a set of full rotations the spiral will have touched everywhere on a radius from the eye with the same coverage if you accumulate the presence" (emphasis added).  Okay, yes, that is something specific to the Air rune... there are other runes that aren't already circular but form a solid disk if you rotate them, but some points in the disk would be passed through during the rotation more times than others.  (The Fate rune also has this property, but it's not a rune commonly associated with mortals.  The Truth rune would have this property if it were drawn with all three legs of equal length, but as it's usually depicted I think one leg is slightly longer.)

(Oh, one more: Like the Truth Rune, the Movement Rune would have this property if all three "arms" were identical... which at first I thought was the case, and I was going to take issue with the claim that it didn't have rotational symmetry.  But on taking a closer look, I think I see what you mean; it would have rotational symmetry if it were a perfect triskelion, but as @scott-martin alludes to it isn't, quite; the bottom two arms join up directly and the top arm joins them at right angles.  This not only destroys the rotational symmetry, but also prevents it from having the equal coverage property of the Air rune.)

[Joerg]

22 hours ago, scott-martin said:

Joerg, I love the note on Storm prefiguring time, how have you been?

This  is all some corrolary my greater strange theory (on Mostal, the World Machine) that I inflicted on Ludo in our latest podcast episode (which should cover at least part of my recent whereabouts).

So, the weird theory is that the World Machine really is a self-replicating engine seeding the Void with a universe, then growing up a set of daughter universes to see the Void elsewhere. Eight ancient castes for the primal matter (Rock), the first four undifferentiated elements (Darkness, Water, Earth, Sky with their respective metals lead, quicksilver, copper and tin) and the three differentiated forms from Sky (brass aka alloyed copper and tin, silver, gold).

Somehow, the brass step was polluted with Disorder, allowing the Grower to override the replication protocol, diverting Creation from the Chaosium into the pregnancy of Earth, stunting the growth of the Cosmic Eggs (IMO eight distributed evenly around the Spike), and ending it entirely when the eruption of Storm sent the World Machine and its protocols akilter.

Minor other problems cascaded out  of control, too, resulting in the Greater Darkness and the Restart In Time.

There are no records for the other seven pseudocosmic eggs, but the one found by the Feldichi got experimented with by the Second Council, and its programming overlapped with the re-written routines of the still functional parts of the World Machine and caused a short re-lapse into the Golden Age cyclical stasis until the new programming took over again. With the Young Elementals escaped from the egg, further replication of this World Machine seems unlikely.

[mfbrandi]

16 hours ago, Jex said:

I'll give it a try, though but I'll put it behind spoiler tags because it may get a bit long and it's not very interesting.

Thanks.

Spoiler

After I had posted, I “figured” — well thought, rightly or wrongly — if you stuck a pin in the paper on which your rune was drawn, stuck a pen on the point on the rune furthest from the pin, joined the two with string, and drew a circle, the circle would completely contain the rune. I hope that is right, and that it holds for any shape (“rune”) and any pin point, whether it be the centre of the rune or not. I wouldn’t trust my mathematical intuition further than I could throw it.

WHERE I WENT WRONG:

Spoiler

I think the shapes I was getting were an artefact of Inkscape’s alignment routines: Inkscape’s centring a figure on the horizontal axis (i.e. vertically) seems to be making the distance from the horizontal axis to the top of the figure the same as the distance from the horizontal axis to the bottom of the figure — which is often what one would want — but of course, the centre of a figure need not be equidistant from its top and bottom, so the results are not the same as aligning the figure’s centre to a horizontal axis.

But that means that the circle drawn using the centre-to-furthest-point method will sometimes be larger than a non-circular shape just big enough to hold the figure in every rotation, right? (Not always: the figure may itself be a circle.) The intuition comes from the fact that the red figure above is a Reuleaux triangle — one of those annoying shapes that refuses to be a circle but has a constant width — which makes a great roller, but a terrible wheel: where would you put the axle?

I reckon — gulp! — that the diameter of a circle needed to enclose a Reuleaux triangle should scale linearly with the triangles width and is about 1.15 × the width of the R. triangle. Clearly, an R. triangle of width 1 would fit inside a 1 × 1 square no matter what its orientation. (In fact, you only need a square with rounded corners, which must have a smaller area.) The area of our enclosing circle must be about π × 0.575 × 0.575, or 1.039.

So the really dumb conjecture — just vague handwaving, really — is that if you superimpose a plane figure in all its orientations and align on the horizontal and vertical axes (Inkscape style), you get the minimum area that will contain the original figure in any orientation. That is not proved by me and may be well known to be false. Obviously, it will contain the figure in all orientations, the question is whether something smaller can. We know the constructed shape is not the same for all figures, as for a circle it is a circle and for a Reuleaux triangle it is a rounded-cornered square.

Edited June 9, 2023 by mfbrandi
tidied up the verbiage

[AlHazred]

4 minutes ago, mfbrandi said:

Thanks. After I had posted, I “figured” — well thought, rightly or wrongly — if you stuck a pin in the paper on which your rune was drawn, stuck a pen on the point on the rune furthest from the pin, joined the two with string, and drew a circle, the circle would completely contain the rune. I hope that is right, and that it holds for any shape (“rune”) and any pin point, whether it be the centre of the rune or not. I wouldn’t trust my mathematical intuition further than I could throw it.

WHERE I WENT WRONG: I think the shapes I was getting were an artefact of Inkscape’s alignment routines:

Inkscape’s centring a figure on the horizontal axis (i.e. vertically) seems to be making the distance from the horizontal axis to the top of the figure the same as the distance from the horizontal axis to the bottom of the figure — which is often what one would want — but of course, the centre of a figure need not be equidistant from its top and bottom, so the results are not the same as aligning the figure’s centre to a horizontal axis.

But that means that the circle drawn using the centre-to-furthest-point method will sometimes be larger than a non-circular shape just big enough to hold the figure in every rotation, right? (Not always: the figure may itself be a circle.) The intuition comes from the fact that the red figure above is a Reuleaux triangle — one of those annoying shapes that refuses to be a circle but has a constant width — which makes a great roller, but a terrible wheel: where would you put the axle?

I reckon — gulp! — that the diameter of a circle needed to enclose a Reuleaux triangle should scale linearly with the triangles width and is about 1.15 × the width of the R. triangle. Clearly, an R. triangle of width 1 would fit inside a 1 × 1 square no matter what its orientation. (In fact, you only need a square with rounded corners, which must have a smaller area.) The area of our enclosing circle must be about π × 0.575 × 0.575, or 1.039.

So the really dumb hypothesis — you cannot call it a theory — is that if you superimpose a plane figure in all its orientations and align on the horizontal and vertical axes (Inkscape style), you get the minimum area that will contain the original figure in any orientation. That is not proved by me and may be false for all I know. Obviously, it will contain the figure in all orientations, the question is whether something smaller can. We know the constructed shape is not the same for all figures, as for a circle it is a circle and for an R. triangle it is a rounded-cornered square.

Is... is this the Nysalorian riddle for Inkscape skill?

[mfbrandi]

2 minutes ago, AlHazred said:

Is... is this the Nysalorian riddle for Inkscape skill?

I certainly hope so.

[mfbrandi]

Yelm is one of the Greater Gods and a key upholder of the Cosmic Compromise. As the Ruler of the Universe, Yelm must interact with the other forces of the cosmos. This includes the gross and unclean material elements of the cosmos …

Yelm is Illuminated and can deal with the polluted world without becom[ing] corrupt through his supreme moral virtue. This includes Chaos of course, for Chaos is part of the universe. But because we are Yelm we cannot be tempted by Wakboth.

— Jeff Richard, Illumination and the Solar Religion

——————————————————————————

But this is a mistake. A mistake only those dazzled by the ruling class would make. The pride is there and the middle way, but Yelm is not the ruler of the universe — the very idea! Yelm is Philip Marlowe:

——————————————————————————

It is not a fragrant world, but it is the world you live in … It is not funny that a man should be killed, but it is sometimes funny that he should be killed for so little, and that his death should be the coin of what we call civilization …

But down these mean streets a man must go who is not himself mean, who is neither tarnished nor afraid. The detective in this kind of story must be such a man. He is the hero; he is everything. He must be a complete man and a common man and yet an unusual man. He must be, to use a rather weathered phrase, a man of honour — by instinct, by inevitability, without thought of it, and certainly without saying it. He must be the best man in his world and a good enough man for any world … he is neither a eunuch nor a satyr … if he is a man of honour in one thing, he is that in all things.

He is a relatively poor man, or he would not be a detective at all. He is a common man or he could not go among common people; he has a sense of character, or he would not know his job. He will take no man’s money dishonestly and no man’s insolence without a due and dispassionate revenge; he is a lonely man and his pride is that you will treat him as a proud man or be very sorry you ever saw him.

— Raymond Chandler, The Simple Art of Murder

——————————————————————————

Breeze looked at me very steadily. Then he sighed. Then he picked the glass up and tasted it and sighed again and shook his head sideways with a half smile; the way a man does when you give him a drink and he needs it very badly and it is just right and the first swallow is like a peek into a cleaner, sunnier, brighter world.

— Raymond Chandler, The High Window

Edited June 9, 2023 by mfbrandi
added extra Chandler quote: alcoholic illumination

[Jex]

7 hours ago, mfbrandi said:

I think the shapes I was getting were an artefact of Inkscape’s alignment routines

  Yes, I think that's correct; it's the "centering" of the shapes that led to the overall shape of your overlaid rotated versions being squarish instead of circular.  (This isn't something unique to Inkscape; I think Adobe Illustrator and most other graphics programs use similar methods of alignment.)

7 hours ago, mfbrandi said:

So the really dumb conjecture — just vague handwaving, really — is that if you superimpose a plane figure in all its orientations and align on the horizontal and vertical axes (Inkscape style), you get the minimum area that will contain the original figure in any orientation. That is not proved by me and may be well known to be false.

That's an interesting conjecture, but as it turns out it is false.  Let's take maybe the simplest noncircular shape possible: a straight line of length D.  Superimpose its aligned rotations, and you get a circle with radius D/2.  This is not, however, the smallest area that will contain this line in any rotation.  The "concave triangle" shape below (technically called a deltoid) also works and has half the area of the circle:

(In fact, for some time it was believed that this is the smallest such shape that fits these criteria... but it was later proven that not only is this not the shape with minimum area, but there is no shape with minimum area; assuming the line is infinitesimally thin, it's possible to get the area as small as you want with an appropriately designed shape (well, as small a positive area as you want; it can't actually get to zero).  If this seems counterintuitive to you, yeah, it's counterintuitive to me too; math can be weird.  I first read about this in the book The Unexpected Hanging and Other Mathematical Diversions, by Martin Gardner; you can read more about it on this Wikipedia page (which is where I got the image above), though it's kind of technical.)

[EDIT: Okay, glancing over that Wikipedia page myself, I find that I was mistaken about one thing in the previous paragraph.  The area can't actually get to zero only if you add the criterion that you must be able to continuously rotate the line segment within the shape.  If you only care that the line segment fits within the shape at any angle and don't care that you can actually rotate it from one position to another without leaving the shape, then the area actually can get to zero!  That... seems really counterintuitive to me, and I have a hard time seeing how that could work (some kind of fractal, maybe?), but apparently it's been proven.  Like I said, math can be weird.]

Edited June 9, 2023 by Jex
Correction to an apparently erroneous statement in the last paragraph

[mfbrandi]

11 hours ago, Jex said:

If you only care that the line segment fits within the shape at any angle and don't care that you can actually rotate it from one position to another without leaving the shape, then the area actually can get to zero!  That... seems really counterintuitive to me, and I have a hard time seeing how that could work (some kind of fractal, maybe?), but apparently it's been proven.

Ha! Ha! My intuitions often trip me up in the other direction.

My first thought would be that each of those one-dimensional lines has area zero; if you combine them, the resulting figure has area zero; you can do this with as many lines as you like, and the area doesn’t go up; equally, no matter how many you add, you cannot manoeuvre a line from one orientation to another — you cannot build a filled-in circle from superimposed zero-area line segments, even if you have uncountably many of them (presumably the possible orientations of a line in a plane is given by the real numbers). THEN I would think that I know shit about infinities, screw up my piece of paper, and throw it away. (That last at least is sound thinking: whatever I think, something else will be true.) 😉

I wasn’t clear how the Kakeya needle thing generalised to 2D shapes, but it seems clear that the concave triangle thing will work for thin rectangles as well as lines, so it is sufficient to trash my really dumb conjecture.

[scott-martin]

16 hours ago, Jex said:

some kind of fractal, maybe?

I can see this operating like an antifractal in that the area enclosed approaches and then achieves a theoretical limit of zero as the line itself is successively stripped (or "tapped") from its second dimension. Death by infinite cuts. We would know this because the region of the circle left out of the rotation would be that fractal section (area = 1) and 1 - 1 would leave zero for the inscribed figure. QED baby. Or something.

The perimeter might be neurologically resonant in some way like how a tattva will flip to its complementary color (or from "convex" to "concave") when you meditate on it hard enough. In general I would like to see advanced sorcery incorporate this kind of technical insight into its training narratives to make them more complex and more fun for those who enjoy . . . and if they didn't enjoy they wouldn't be playing sorcerers!  

The moral: every rune inverts depending on whether you're experiencing it from "the inside" or not. This is how polarity works in the powers and how the elements contain a reversible sequence. There are other tricks. The side with consciousness in it is the fractal side, hogging all the air (or whatever) from the situation.

[mfbrandi]

2 hours ago, scott-martin said:

this kind of technical insight into its training narratives to make them more complex and more fun for those who enjoy . . . and if they didn't enjoy they wouldn't be playing sorcerers! 

What was that Larry Niven story, Convergent Series? IIRC, the sorcerer deals with a troublesome demon by marking out a pentacle on the demon and then conjuring it to appear within the pentacle.

But I may misremember.