Two pairs of ants set off from the same starting point, walking side-by-side. One pair takes the red path, the other pair takes the blue path.
The red and blue paths are geodesics (the straightest possible paths, like great circles on a sphere), and each ant in a pair follows a geodesic that lies a small distance to either side of the marked one, initially pointing in the same direction.
If this ellipsoid has semi-axes of length 1 and 3, the ants taking the blue path will bump into each other at the “pole”, where all meridians converge, after travelling a distance of 3.34122.
How far will the ants following the red path travel before they bump into each other? More, less, the same?
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Lying and politics: a brief primer
Dr Philip Seargeant looks at the affectionate relationship between lying and politics.Open Learning
Democrat Taylor Rehmet wins a reliably Republican Texas state Senate seat, stunning GOP
https://www.npr.org/2026/02/01/nx-s1-5695678/democrat-taylor-rehmet-wins-texas-state-senate-seat?utm_source=flipboard&utm_medium=activitypub
Posted into U.S. News @u-s-news-npr
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RE: bsky.app/profile/did:plc:elbx2…
Trump says feds won't intervene during protests in Democratic-led cities unless asked to do so
https://www.npr.org/2026/02/01/nx-s1-5695677/trump-feds-protests-democratic-cities?utm_source=flipboard&utm_medium=activitypub
Posted into U.S. News @u-s-news-npr
Journalist Claire Keenan takes up ocean swimming for the first time. ‘I’ve swum and squealed alongside enormous blue gropers, tiny stingrays, silver and striped fish and even a Port Jackson shark,’ she writes.
Photograph: Jessica Hromas/The Guardian
#photography
#ocean
#swimming
#women
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Right-wing #nationalism is a #bourgeois nationalism, and in our struggles against #capitalist austerity we must emphasize that our #enemy arrives in a limousine, and not on a boat.
Harsha Walia, Border and Rule: Global Migration, Capitalism, and the Rise of Racist Nationalism
"France installs 5.9 GW of PV in 2025"
#France #SolarPower #Energy #Renewables
pv-magazine.com/2026/01/26/fra…
France installs 5.9 GW of PV in 2025
The country's cumulative installed PV capacity reached 31.1 GW at the end of December.pv magazine International
Jet and I had travelled 4,000-plus kilometres from Sydney in two-and-a-half weeks. We both needed a break. But first, I wanted to see Uluṟu again. ❤️🐶🚍
Link to post on Tall And True: tallandtrue.com.au/blog/a-dog-…
#dogs #vanlife #roadtrips #sydney #uluru
A Dog and Van Road Trip - Part Two: Uluru
A Dog and Van Road Trip - Part Two covers the six days I spent at Uluru with my 14-year-old black Labrador, Jet.Tall And True
The only thing worse than websites that auto-play videos are websites that auto-pause the video the second you switch tabs.
I wanted background noise, not separation anxiety 🙄
Or that pause when you scroll down.
Pretty sure imgur and bsky both do this and it's SO annoying.
fanpage.it/esteri/in-ucraina-d…
Fanpage enfatizza il “fallimento strategico della Russia”.
Adnkronos invece parla del rischio di cedimento delle difese ucraine: “Dopo 4 anni di guerra, l'Ucraina rischia di cedere in più punti. La difesa di Kiev, costretta a coprire un fronte che si estende per oltre 1100 km, fatica a opporsi ad ogni offensiva della Russia. Mancano i soldati per contrastare ogni azione del nemico”.
Guatemala Opts Out of Oil Extraction in Favor of Protecting Jaguars and Macaws in Mayan Biosphere
<p>Guatemala has opted out of renewing a lease agreement on a 7,000-acre oil field in order to use the land for better protection of the surrounding L
🔗 goodnewsnetwork.org/guatemala-…
Guatemala Opts Out of Oil Extraction in Favor of Protecting Jaguars and Macaws in Mayan Biosphere
Natural Resources Minister Patricia Orantes said the government is trying to do a “180-degree” shift on many management decisions.Andy Corbley (Good News Network)
Sheep run through a paddock as bushfire smoke rises on nearby hills at Longwood, Victoria. Across the region houses were reduced to warped steel, with brick chimneys often the only thing left standing.
Photograph: Steve Womersley/The Guardian
#photography
#Australia
#bushfires
#wildfires
#sheep
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Rencontre avec Christelle Taraud et Emmanuelle Walter autour de leurs livres sur les féminicides
Rencontre croisée jeudi 12 février au Monte en l’air avec Christelle Taraud & Emmanuelle Walter sur leurs livres Les Filles-au-Diable et Sœurs volées
Rencontre avec Christelle Taraud et Emmanuelle Walter autour de leurs livres sur les féminicides
Rencontre croisée jeudi 12 février au Monte en l'air avec Christelle Taraud & Emmanuelle Walter sur leurs livres Les Filles-au-Diable et Sœurs voléesParis-luttes.info
Lotta # 1 – Français
Premier Numéro en français d’un bulletin multilingue anarchiste communiste, antimilitariste, défaitiste révolutionnaire et pro-lutte de classe.
Lotta # 1 – Français
Premier Numéro en français d'un bulletin multilingue anarchiste communiste, antimilitariste, défaitiste révolutionnaire et pro-lutte de classe.Paris-luttes.info
Around eighteen months after starting, my wife's completed the second of her king cards cross stitches, the King of Hearts. Like the first one this one is about 110,000 stitches. Two to go.
Looks a bit like Star Trek's Q to us.
#CrossStitch #XStitch #Craft #Crafts #StarTrek
Greg Egan
in reply to Greg Egan • • •To answer this, we can start with the fact that on a *sphere* of radius R, any two ants following neighbouring great circles would travel a distance of (π/2) R before their paths met. That behaviour comes from the intrinsic curvature of the sphere, which is measured by its Gaussian curvature, K = 1/R^2.
But what is the Gaussian curvature of our ellipsoid, at its equator? The equator itself has a curvature of 1, but the meridians are ellipses that are 3 times wider. What is the curvature of an ellipse with semi-axes a and b, at the point where the minor semi-axis meets the ellipse?
The ellipse takes the form:
y(x) = b √[1-(x/a)^2]
For small x:
y(x) ≈ b (1-½(x/a)^2)
The second derivative is:
y''(x) = -b/a^2
If we set a = b = R this would be -1/R, so the curvature here is just the opposite of the second derivative.
The Gaussian curvature is the product of the curvatures along two perpendicular directions where it achieves its maximum and minimum values. For our ellipsoid, these curvatures are 1/b for the equa
... show moreTo answer this, we can start with the fact that on a *sphere* of radius R, any two ants following neighbouring great circles would travel a distance of (π/2) R before their paths met. That behaviour comes from the intrinsic curvature of the sphere, which is measured by its Gaussian curvature, K = 1/R^2.
But what is the Gaussian curvature of our ellipsoid, at its equator? The equator itself has a curvature of 1, but the meridians are ellipses that are 3 times wider. What is the curvature of an ellipse with semi-axes a and b, at the point where the minor semi-axis meets the ellipse?
The ellipse takes the form:
y(x) = b √[1-(x/a)^2]
For small x:
y(x) ≈ b (1-½(x/a)^2)
The second derivative is:
y''(x) = -b/a^2
If we set a = b = R this would be -1/R, so the curvature here is just the opposite of the second derivative.
The Gaussian curvature is the product of the curvatures along two perpendicular directions where it achieves its maximum and minimum values. For our ellipsoid, these curvatures are 1/b for the equator and b/a^2 for the meridian, so we have:
K = 1/a^2
In other words, the Gaussian curvature at the equator of the ellipsoid is completely independent of b (the radius of the equator) and is the same as that of a sphere of radius a (the polar radius)!
That in turn means that the ants following the red path will bump into each other after travelling a distance:
(π/2) a = 3π/2 ≈ 4.71239
So, perhaps a bit counterintuitively, the ants following the small red circle will travel *farther* before they converge than the ants following the larger blue ellipse.
Rahul Narain
in reply to Greg Egan • • •foo ✅
in reply to Greg Egan • • •soaproot
in reply to foo ✅ • • •Erin
in reply to soaproot • • •Greg Egan
in reply to Erin • • •@ChateauErin @soaproot @slashdottir
For the blue path, which is a meridian, the ants that walk side-by-side and “follow the blue path” just walk along two meridians with longitudes that are a tiny bit west, and a tiny bit east, of the one that is drawn. All three meridians meet at the pole, so the ants bump into each other at the pole.
For the red path, again the ants walk side-by-side, each of them a very small distance on either side of the red path (i.e. one of them very slightly north of the equator, the other one very slightly south of the equator). But circles of latitude are not geodesics, so their latitudes do not stay constant.
I did not draw the individual paths for the ants because they're meant to be an “infinitesimal" distance on either side of the paths that are drawn.
Erin
in reply to Greg Egan • • •@soaproot @slashdottir hmm. If the red paths are so close to the red circle (which is of radius 1, right?) as to be indistinguishable, I don't understand how the path length is so different from the (half? since the paths meet at ascending and descending nodes?) circumference of the unit circle. Except that that must be the point of the exercise, but if it's so visually similar...hm.
Sorry for being the thick student but I get concerned when I don't understand ellipses
Greg Egan
in reply to Erin • • •@ChateauErin @soaproot @slashdottir
On a sphere of radius 1 the ants following the red path would meet after they have gone 90 degrees around the equator, so one quarter of the circumference. If you extended their tracks into complete great circles, the distance between their two crossing points would be half the circumference.
On the ellipsoid, their paths still stay very close to the equator (the absolute value of their latitude is a maximum when they start, and I have said that is very small). But being close to the same circle, on a very different surface, doesn't mean they should meet up after a similar distance.
For example, if the red circle was on a cylinder, the ants would never bump into each other.
Erin
in reply to Greg Egan • • •@soaproot @slashdottir ohhhh. I somehow hadn't understood they were starting at the latitude extrema, but I see now why that's the case. hm.
I may need some kind of animation to understand the path the ants are taking and why they're not meeting where I expect / where the geodesics intersect. But obvs not your job to provide it
Greg Egan
in reply to Greg Egan • • •Lien Rag
in reply to Greg Egan • • •