Elversprog, klingonsk og ‘Pig Latin’: Hvad kan opfundne sprog lære os om den menneskelige hjerne?

This week the #puzzle is: How Far Can You Roll? #arclength #estimate #coding #montecarlo #average

For her photography show, Frederica Fiddleria attaches a light to a point on the circumference of a circular wheel with a radius of 1 meter. She points a camera at the wheel and, during a single long exposure, rolls the wheel for one revolution along the ground.

When she develops the film, she is curious about the path the light took as the wheel rolled. What is the length of this path?

And for extra credit:

For her next show, Frederica wants to mix things up. Instead of placing the light on the circumference of the wheel, she will pick a random point inside the circle. (Before you ask, let me clarify what “random” means here: Any two regions with the same area are equally likely to contain the point.)

As before, she will roll the wheel for one revolution along the ground and capture the motion with a single long exposure on her camera. On average, what can she expect the length of the path to be?

How Far Can You Roll?

Intermission

Last week I didn’t really try to solve the fiddler. It involved learning rules to games, solving a chess puzzle, and picking up on subtle verbal cues. None of them are my strong suits.

Solution, possibly incorrect:

Cycloid

Okay, so this will be a little stream of consciousnessy. Apparently the curve, we’re looking for is a cycloid. And the length of the curve is an arc length. For a cycloid, the arc length is $8r$ (see source above). With $r=1$, the arc length will be $8\cdot1=8$.

Result: 8 meters. (Method 1.)

Oh. And I assume the distance between camera and light is large enough, that I don’t have to account for whatever it’s called, when the motive is distorted. Perspective? Parallax?

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In general the length of a path (the arc length) is:

Just to test how this works, let’s look at a circle.

And combining all this:

This fits what we expect. So I guess I know how to use this formula.

For the cycloid, the formulas are:

Or how WolframAlpha would put it:

integral square root((1-cos(t))^2+sin(t)^2) from 0 to 2pi

Everything seems to work okay. I guess that was method 2. Moving on.

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And for extra credit:

Trochoid Arc length of a trochoid Program

This time the curve is a trochoid. A cycloid as a special case of that. And it’s my lucky day! The arc length of that one is already calculated Out There. A new parameter is b (again), the distance between the point on the circle and the center of the circle.

Where E is an elliptical integral. Let me see… Ah yes.

Hm. There’s something wrong. E can’t have both 1 and 2 parameters.

There’s another definition of L:

I think I’m supposed to read it like this actually:

And then combining:

No. Scratch method 1. Let me go back to the definitions for the point.

Then I code a monte carlo approach, where the arc length is estimated by assuming I can just look at some points along the curve and calculate at all the little straight distances between these. Just to make sure my approach works, I also code the fiddler and sort of get the same result. Then I run the program a few times.

Result 2: 7.109200
Result 2: 7.112045
Result 2: 7.111690
Result 2: 7.111431

I am confident I can say the rounded answer is 7.11. Method 2 worked out. And I guess also gave a method 3 for the fiddler.

This week the #puzzle is: How Lucky Can a Baseball Team Get? #probabilities #counting #montecarlo #coding

The Fiddler Baseball League consists of exactly two teams of equal skill: the Algebraists and the Geometers. Over the course of a season, these two teams play each other 162 times. Each team has an equal chance of winning each game, and the results of games are independent of one another.

At the end of the season, on average, how many games would you expect the team with the better record to have won? (If the teams have the same record, then you should include one of them in your calculation.)

And for extra credit:

After some expansion, the Fiddler Baseball League now boasts 30 teams. Over the course of a season, each team plays each other team five times. (Thus, each team plays a total of 145 games.) As before, each team has an equal chance of winning each game, and the results of games are independent of one another.

At the end of the season, on average, how many games would you expect the team with the best record to have won? (If more than one team has the same best record, then you should include one of them in your calculation.)

How Lucky Can a Baseball Team Get?

Solution, possibly incorrect:

Program

Method 1: Monte Carlo. Play through a tournament of 162 games. Do this in a loop a lot of times. Run the program a few times to find a stable result.

Method 2: Calculate the average directly. E.g., with 82 wins, there’s a probability of $\frac{{162 \choose 82} }{2^{162}}$ of this situation arising. Multiply by 82. Sum over all win options, 0-162. Note that 82 wins and 80 wins both produce “winner won 82 times” situations.

Result: 86.06988.

And for extra credit:

Method 1: Definitely Monte Carlo.

Result: $\underline{84.98 \pm 0.01}$.

Anmeldelse af “best related”. 2025. Hugo-finalist.

Nebula nominees . Hugo finalists .

Colourfields: Writing About Writing About Science Fiction by Paul Kincaid

Skitse: En samling af anmeldelser og andre essays, der har med sf at gøre.

Er det science fiction? Absolut.

Er det godt? Ja, det er det vel egentlig. Velskrevet og interessante holdninger. Så hvis jeg virkelig godt kunne lide anmeldelser … ##-

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The Hugo Spreadsheet of Doom, maintained by Renay

Skitse: En liste over værker af forskellig art, der kan inspirere til nominering til hugoen.

Er det science fiction? Sff osv.

Er det godt? Sådan noget som det her et regneark, men det er vel også arbejdet med at putte indhold i det, administrere det, tage stilling til om det enkelte værk egentlig kan nomineres osv. Tja. Godt arbejde. Men det føles også en anelse meta at have det med på stemmesedlen. ##-

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Inventing the Renaissance by Ada Palmer

Skitse: Hvordan og hvorfor opstod begrebet renæssance? Fyldig historie.

Er det science fiction? Nix. Historie. Men Palmer har skrevet sf ved andre lejligheder.

Er det godt? Det virker velskrevet. Det virker bare også meget underligt at have det på stemmesedlen. Jeg tror, jeg vil undlade at give det min stemme på nogen måde. Dvs., jeg vil ikke placere det under “no award”, blot ikke stemme på det. #–

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Last War in Albion: “The Cuddled Little Vice (Sandman)” by Elizabeth Sandifer

Skitse: En gennemgang af ham der, der skrev Sandman-bøgerne. Hvem er han, hvordan opstod tegneserierne, og ikke mindst, er han et godt menneske?

Er det science fiction? Eftersom Sandman mere er fantasy (og horror?), så nej.

Er det godt? Jeg læste den her, da den var ny, og gik meget op i det. Nu er det hele sådan lidt blah. Endnu en fyr, der virker ret skummel. Suk. ##-

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Positive Obsession: The Life and Times of Octavia E. Butler by Susana M. Morris

Skitse: Biografi om Butler, inklusive hvad der i øvrigt foregik i verden på de relevante tidspunkter, og hvordan Morris selv passer ind i historien.

Er det science fiction? Jeps, eftersom Butler mest skrev sf.

“A war waged with world-destroying weapons would be the pinnacle of the Human Contradiction, an example of both our stunning intelligence and our self-destructive hierarchal thinking.”

Er det godt? Ja. Der er lidt gentagelser hist og her, som om den sidste revision ikke bed helt nok. Men det meste er bare velskrevet og interessant. ###

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“Ragnarök vs the Long Night” by Ashaya and Aziz

Skitse: Et afsnit af en podcast, der tager udgangspunkt i Game of Thrones/A Song of Ice and Fire. I lige det her afsnit skal vi diskutere ragnarok, fordi Martin har ladet sig inspirere af dette fænomen.

Er det science fiction? Nix.

Er det godt? Hvis man er fan af serien, hvis man kan goutere forkert udtale af Yggdrasil og Tolkien, hvis man kan håndtere 10 x “vi kommer tilbage til” på stribe, hvis man kan håndtere entusiasme (der i hvert fald på mig virker falsk), så kan man nok godt lide den her podcast. ##-

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Butler-bogen tager ubesværet førstepladsen.

This week the #puzzle is: Can You Sneak Past the Rings? #probabilities #montecarlo #coding #plot

A long vertical cylinder has three narrow open rings, each of which wraps around seven-eighths of the cylinder (leaving a one-eighth “gap”). The rings are evenly spaced vertically, but are otherwise randomly rotated about the cylinder’s central axis. For some orientations of the rings, there exists at least one vertical line down the cylinder’s surface that passes through each ring’s gap, as illustrated below.

What is the probability that at least one such vertical line exists?

And for extra credit:

Instead of requiring a vertical line down the cylinder’s surface, now any helix down the surface is allowed. An example of such a helix passing through all three gaps is shown below.

What is the probability that there exists at least one such helix that can pass through each ring’s gap?

Can You Sneak Past the Rings?

Solution, possibly incorrect:

Program Desmos

Method 1: Monte Carlo. It’s a bit fiddly, because when exactly do the 3 gaps overlap? I settle on:

• Put the 1st gap at 0 – 1/8, WLOG.
• If possible, create the overlap between the 2 1st gaps.
• Then look for an overlap between that overlap and the 3rd gap.

467910 sneaks out of 10000000, prob = 0.04679
468490 sneaks out of 10000000, prob = 0.04685
468187 sneaks out of 10000000, prob = 0.04682

Once I had all those conditions in place, I could actually do method 2, plot it in Desmos.

It was fun to figure out how to plot it. How the cases combine. I look at the cases for the double overlap in the columns and the cases for the triple overlap in the rows.

Gap 1: a-b, 0 – 1/8 Gap 2: c-d, c = x, $0\le x\le1$ Gap 3: e-f, e = y, $0\le y\le1$ In general, overlap goes $i_1 – i_2$	Overlap: c-b, x – 1/8 $0\le x\le\frac{1}{8}$	Overlap: a-d, 0 – [x – 1 + 1/8] $\frac{7}{8}\le x\le1$
$i_1\le e\le i_2\iff i_1\le y \le i_2$	$x\le y\le\frac{1}{8}$	$0\le y\le x-1+\frac{1}{8}$
$i_1\le f\le i_2\iff i_1\le y-1+\frac{1}{8}\le i_2$	$x\le y-1+\frac{1}{8}\le \frac{1}{8}$	$0\le y-1+\frac{1}{8}\le x-1+\frac{1}{8}$
$e\le f$ $e\le i_1,i_2\le f\iff y\le i_1,i_2\le y+\frac{1}{8}$	$y\le x$ $\frac{1}{8}\le y+\frac{1}{8}$	Impossible
$f \le e$ $0\le i_1,i_2\le f \iff i_2\le y+\frac{7}{8}$	Impossible	$\frac{7}{8}\le y$ $x-\frac{7}{8}\le y – \frac{7}{8}$

Anyway. The result is 6 triangles. Each triangle has an area of $\frac{1}{8}\cdot\frac{1}{8}\cdot\frac{1}{2}=\frac{1}{128}$. The area of all the triangles is $\frac{6}{128}=\frac{3}{64}=0.046875$. This agrees with method 1.

Result: 0.046875.

And for extra credit:

Desmos

I recreated the situation in Desmos and played around a little. Among other things, I construct the (orange) interval e has to fall in, in order to make a helix possible.

Interestingly, this interval always has length 0.5, corresponding to a probability of 0.5.

• My x axis shows how far along the circumference of the cylinder I am. As with the fiddler, the first gap is frozen 0 – 1/8.
• The helix has to go through a and d, or b and c, or somewhere in between. This corresponds to different slopes of the helix.
• I construct a point on the line going through b and c and crossing the 3rd ring. This point will have $x=c+(c-b)=2c-b$.
• This leads to $2c-b\le f\iff 2c-b-\frac{1}{8}\le e$.
• I also construct a point on the line going through a and d and crossing the 3rd ring. This point will have $x=d+(d-a)=2d-a$.
• This leads to $e\le 2d-a$.
• The interval for e to fall within is therefore $2c-b-\frac{1}{8}\le e\le 2d-a$.
• The length of this interval is
  • $2d-a-(2c-b-\frac{1}{8})$
  • $2d-a-2c+b+\frac{1}{8}$
  • $2d-2c+b-a+\frac{1}{8}$
  • $2(d-c)+(b-a)+\frac{1}{8}$
  • $2\cdot \frac{1}{8} +\frac{1}{8}+\frac{1}{8}$
  • $\frac{1}{2}$

So the result is 0.5. I am surprised, that the answer is so simple.

Her er årets nominerede noveller, med samt link til mine anmeldelser:

“In My Country” by Thomas Ha	##-	Nebula/Hugo-noveller
“Laser Eyes Ain’t Everything” by Effie Seiberg	##-	Nebula/Hugo-noveller
“Missing Helen” by Tia Tashiro	###	Hugo-noveller
“Six People to Revise You” by J.R. Dawson	###	Nebula/Hugo-noveller
“10 Visions of the Future; or, Self-Care for the End of Days” by Samantha Mills	###	Hugo-noveller
“Wire Mother” by Isabel J. Kim	###	Hugo-noveller

Den var værre. 4 topscorere. Helen og revisionerne er i top 2. Hm. Det kan jeg ikke bestemme mig for nærmere.

Her er årets nominerede noveller, med samt link til mine anmeldelser:

“Through the Machine” by P.A. Cornell	##-	Nebula-noveller
“Six People to Revise You” by J.R. Dawson	###	Nebula/Hugo-noveller
“In My Country” by Thomas Ha	##-	Nebula/Hugo-noveller
“The Tawlish Island Songbook of the Dead” by E.M. Linden	###	Nebula-noveller
“Because I Held His Name Like a Key” by Aimee Ogden	##-	Nebula-noveller
“Laser Eyes Ain’t Everything” by Effie Seiberg	##-	Nebula/Hugo-noveller

Blandt 2 topscorere, så er det alligevel revisions-historien, der løber med guldet.

Anmeldelse af 3 noveller. 2025. Hugo-finalist.

Nebula nominees . Hugo finalists .

“Missing Helen” by Tia Tashiro

Skitse: Helen er ikke længere sammen med Mark, og nu meddeler Mark, at han skal giftes med Helens klon. Ups.

Er det science fiction? Oh yes.

Temaer: Hvor skal man begynde? At folk kan sælge deres DNA, så der kan fremstilles en klon. Omend der er noget lovgivning omkring det, så det ikke bliver alt for vildt. Og det er også smart ikke at acceptere auktionens første bud. Og der er tale om store penge, så det kan også betyde rigtig meget, hvis man har små penge.

At Helen og Mark stadig er venner. Men vist på den der lidt akavede måde … Det sluttede ikke så godt, Helen var en anelse omklamrende.

Teksten er i 2. person. “Du” = Helen.

“When Mark told you he was marrying your clone, you said “congratulations” on autopilot, because that’s what you say when your friend tells you they’re getting married, and you had promised to stay friendly after the divorce.”

Er det godt? Ja! Den her var tilfredsstillende. ###

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“10 Visions of the Future; or, Self-Care for the End of Days” by Samantha Mills

Skitse: Der er muligvis nogen, der vil bygge en port til Helvede i nærheden af California. Det lykkes muligvis for dem. Det har muligvis drastiske konsekvenser.

Er det science fiction? Nja, en port til Helvede kan vel ikke kategoriseres på den måde. Så fantasy.

Temaer: Teksten har 10 dele, der viser hver sin version af fremtiden. Fortælleren har set disse visioner. Det er rart at vide senere, når man selv er landet i en skidt fremtid, at det kunne have været værre.

Uanset hvor grum den aktuelle situation er, uanset porte, grumme, indtrængende væsner osv., så er der dog også en styrke i at være del af et par. Måske ovenikøbet noget sjovt. Som da parret slap væk, bl.a. med et spil Yatzy. Den slags er vigtigt.

Samtidig med, at meget gammeldags ulykker rammer Jorden, så er der meget moderne referencer. Visionerne er kommet ind via en krystalkugle købt på Amazon.

Er det godt? Jeps. Den er fin. ###

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“Wire Mother” by Isabel J. Kim

Skitse: Cassie har en digital mor. Og det er noget skidt, fordi Cassie har svært ved at relatere ordenligt med digitale personer.

Er det science fiction? Jøp.

Temaer: Det er simpelthen noget skidt at være gammel nok til at have problemer med sine forældre, og så få det her digitale element oveni.

Det er rutine, at en digital person kan leje en kødpersons krop. Således kan far også få noget på den dumme en gang imellem.

Afhængig af alder, så kan hvem som helst skabe en digital person. Selvfølgelig skaber teenagere digitale kærester og lærer de forkerte lektier af det.

“It’s one of the ironies of the modern era: it’s an expensive, optional luxury to create a digital person, but every meat person has a legal obligation to create a biological child or at least manage to sell their mandatory child credit to someone who wants a freakishly large family, instead.”

Er det godt? Nemlig. ###

Anmeldelse af 3 noveller. 2025. Hugo-finalist. Nebula-nomineret.

Nebula nominees . Hugo finalists .

“Six People to Revise You” by J.R. Dawson

Skitse: Liza er i gang med et projekt, hvor de skal bede 6 personer i deres liv om at udfylde en formular. I den ene ende partner, venner, familie, i den anden ende netop en, der ikke er nogle af delene. Formularen giver folk mulighed for at foreslå forbedringer af Liza.

Er det science fiction? Nemlig.

Temaer: Det bliver langsomt rullet op, at projektet er, at Liza skal forbedres, fysisk og psykisk. (Ingen detaljer om hvordan.) Så man kan ønske, at de får mere selvtillid, eller mindre tænder. Alle ønskerne vil blive lagt sammen til et harmonisk hele.

Lizas mor ønsker selvfølgelig, at det der stedords-hejs skal stoppe. Nogle i omgangskredsen ønsker det bedste for Liza, andre virker mere “det ville være godt for mig, hvis Liza blev sødere”.

“Everyone changes through life, even if they don’t have a revision.”

Er det godt? Ja. ###

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“In My Country” by Thomas Ha

Skitse: I princippet er det her en historie om et land. I det her land er der bl.a. blå huse, og i de blå huse er der mænd. Nogle gange står sådan en mand pludselig i stuen eller soveværelset, og det betyder … noget.

Er det science fiction? Mja. Det er det vist.

Temaer: Blå-hus-mænd er del af en større struktur, der via flere led går helt op til Den Første Borger. (Der ikke er konge!) Nogle gange rejser man til De Fjerne Marker, efter en samtale med sådan en mand. Naturligvis på en god måde.

Landet har bl.a. den regel, af fiktion skal være entydigt. Det skal være nemt at finde ud af, hvem der evt. bliver stukket til, og hvilke holdninger forfatteren har. Hovedpersonens søn begynder at skrive historier, der ikke holder sig indenfor stregerne.

Er det godt? Åh ja. Nu har jeg selv hang til historier, der er nemme at dekode. Så både den her novelle og den historie, den har i maven, passer mig ikke så godt. ##-

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“Laser Eyes Ain’t Everything” by Effie Seiberg

Skitse: Fortælleren fik et stykke af et rumskib i hovedet fornylig og har nu laserøjne. Kørestolen er et ældre fænomen. Fagforeningen for superhelte er interesserede i øjnene, men har ikke tænkt sig at installere en rampe. Ingen kan få ind i deres hoveder, at superøjne ikke betyder, at man er fri for handicap.

Er det science fiction? I den grad superhelte er det, ja.

Temaer: Fortælleren går egentlig op i rundkørsler, fordi der sker færre uheld i sådan nogle end i kryds. Er det foreneligt med at være en superhelt?

Er noget som helst overhovedet foreneligt med en fagforening, der tænker mere i PR?

“ExpandiRay wouldn’t lend me his raygun to grow an intersection wider so that I could carve it into a circle after, because insurance wouldn’t cover it.”

Er det godt? Jeg kan godt lide slutningen, men det er ikke helt nok. ##-