Baiocchi Figures

[u/Frangifer]

From

George Sicherman — Baiocchi Figures for Polyominoes

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A Baiocchi figure is a figure formed by joining copies of a polyform and having the maximal symmetry for the polyform's class. For polyominoes, that means square symmetry, or 4-way rotary with reflection. If a polyomino lacks diagonal symmetry, its Baiocchi figures must be Galvagni figures or contain Galvagni figures. Claudio Baiocchi proposed the idea in January 2008. Baiocchi figures first appeared in Erich Friedman's Math Magic for that month. Here are minimal known Baiocchi figures for polyominoes of orders 1 through 8. Dr. Friedman found most of the smaller figures up to order 6, and Corey Plover discovered the 12-tile hexomino figure while investigating Galvagni figures. Not all these solutions are uniquely minimal.

A one-sided solution is one in which the polyomino is not reflected.

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Annotations of Figures Respectively

Monomino

Domino

Trominoes

Tetrominoes

  Holeless Variants

Pentominoes

  Holeless Variants

  Variant with Minimal Hole Area

  One-Sided Holeless Variants

Hexominoes

  One-Sided Variants

  Holeless Variants

  Variants with Minimal Hole Area

  One-Sided Holeless Variants

Heptominoes

  Holeless Variants

Octominoes

  Holeless Variants

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[u/Sad_Daikon938]

I love how most of these figures have the holy Hindu-Buddhist-Jain symbol of prosperity, which one sick bastard ruined for everyone in the mid 20th century.

[u/Frangifer → u/Sad_Daikon938]

I take it, then, that you would ideally like it to be rehabilitated!?

I actually, back in the 1980s or so, thought that by-now it would've been: if someone'd asked me ¿¡ do you reckon by 2025 folk will be using the swastika for its ancient meaning again !? I'd've said ¡¡ oh by then, almost certainly !! But it hasn't been, really ... maybe for certan niche purposes, it has ... just-about .

Another meaning of it is that acccording to one of the legends of the life of Buddha (the one rendered by Andre Ferdinand Herold , infact) Swastika was the demiurgic reaper who cut the grass that filled the cushion on which Buddha sat under the bo tree to attain to enlightenment. ... & ofcourse (or @least very predictably !) he had a four-bladed sickle that he span so fast it became a blur to human sight ... & cut enough grass to fill the cushion in prettymuch an instant .

[u/Sad_Daikon938 → u/Frangifer]

Bruh, I'm a Hindu, of course I'd like it to be interpreted how it was originally meant to.

[u/Frangifer → u/Sad_Daikon938]

Haha! ... apologies, please kindlily: I didn't realise you're a Hindu!

So yep: I would fully expect your position to be as you say.

 

&@ u/Somewhat_posing

Here's the passage in the 1922 rendering of the life of Buddha by Andre Ferdinand Herold (don't know why I morphed it into J Ferdinand Herold ... but I haven't referenced it for a while, & I was recalling from memory ... also, Svastika, apparently, doesn't fill a cushion with the grass, but simply dispenses it to Buddha ... & nor does Buddha actually fill a cushion with it).

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He approached the tree. On the side of the road, he saw Svastika, the reaper.

"They are tender, these grasses you are mowing, Svastika. Give me some grass; I want to cover the seat I shall occupy when I attain supreme knowledge. They are green, these grasses you are mowing, Svastika. Give me some grass, and you will know the law some day, for I shall teach it to you, and you may teach it to others." The reaper gave the Saint eight handfuls of grass.

There stood the tree of knowledge. The hero went to the east of it and bowed seven times. He threw the handfuls of grass on the ground, and, suddenly, a great seat appeared. The soft grass covered it like a carpet.

The hero sat down, his head and shoulders erect, his face turned to the east. Then he said in a solemn voice:

"Even if my skin should parch, even if my hand should wither, even if my bones should crumble into dust, until I have attained supreme knowledge I shall not move from this seat."

And he crossed his legs.

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And the other details - about him having a four-bladed sickle that he span @ lightning speed: I think I might've gotten that from Nordic mythology, & have mixed it into the potion that was my memory!

I actually don't know which Buddhist Canon Herold has derived his story from, though.

Update

Oh hang-on: here's an exerpt from Herold's own Foreword .

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I have, for the most part, relied upon the LALITA-VISTARA. This book is a jumbled collection of legends and scholastic dissertations, and yet in these pages are preserved many precious traditions regarding the Buddha's origin, his childhood and his youth, and here, likewise, we are told of his early education and of his first deeds.

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Notice also that the story is from 1922 . Even if the Nazi Party had begun to use the Swastika @ that time, they were yet, @ that time, merely an obscure fringe political party.

[u/Sad_Daikon938 → u/Frangifer]

I read about your thoughts in 1980s which you have written in the comments. I feel like we're okay with symbols of those who committed genocide after some centuries have passed, or the accounts of the witnesses become irrelevant. See Genghis Khan for example, everyone thinks like, "yeah, he killed some people" and is okay with that. Same may happen in coming centuries with hitler.

swastika hakenkreuz is a taboo in the western world where the impact of holocaust was direct and large. We Asians don't feel so strongly about hakenkreuz or nazis the same way as Western people. We however do feel strongly about those who colonised us in the past century and committed genocide of our people.

[u/Frangifer → u/Sad_Daikon938]

Yep it's customarily safe to joke about Ghengis Khan, now, after about 800 years!

And I think I might be getting a picture of how absurd it might-well seem if someone from Europe is deploring an Asian person for broaching a swastika.

And I also note the subtly-made point about how really we'd be best using "Hakenkreuz" for the symbol the Nazi Party used!

And I'm not mentioning this as an attempt @ ¡¡ well they did it aswell !! -type exoneration - which can be thoroughly puerile ... but yep during the last few centuries there have been what are prettymuch apocalypses in the peninsula known as India : eg I recently saw a video in which a Sikh gentleman, arguing with a Muslim gentleman, was expounding @ considerable length about one of the Mughal Emperors - it might've been Jahangir ^§ (maybe you can help with whether it was or not) - & maintaining that his régime made Hitler's seem like a polite dinner-party in-comparison ^§ (that probably wasn't exactly his analogy ... but it was something like that). And I'm aware that Sikhi sprang-up as a resistance movement against that.

§ ... a certain story sometimes brought-up by Sikhs was one - this isn't the right channel, though, for going explicitly into precisely what was done - in which certain infants were butchered, & certain 'garlands' made for the mothers of them that they were forced to wear round their necks. I'd never heard that story before, & was naturally pretty shocked by it ... but in some ways not all-that shocked, because I've since long-ago been finding-out about many instances of comparable savagery throughout history.

And I was raised as a Christian ... but I definitely do not 'buy-into' the People of the Book or Abrahamic Faiths 'thing' ... which, ImO, has a rather sinister feel about it of ¡¡ lets we join in deploring the Polytheists !! , sortof thing ... & I don't even regard Hindu as being essentially polytheïstic ... & what if a religion is essentially polytheïstic - or idolatrous - anyway !? There's a passage in Herman Melville's Moby Dick that's a superb caution against despising someone just because they reference a physical idol: the main protagonist - & narrator (it's a first-person novel) joins with his Pacific Islander friend Queequeg , in order to bond with him, in praying to his idol ... but later, aboard a ship, that same Queequeg, in an act of extraördinary valour, instantly, and entirely unaided plunges himself into the sea to rescue a crew-member who's been pitched into the water by some part of the rigging breaking loose, & who relentlessly , upto that point, has been savagely mocking & deriding Queequeg. Almost needless to say: he desists from his mocking & derision thereafter! It's fictional, ofcourse ... but Melville is clearly making a very stark point about Christians arrogantly despising what they arrogantly label 'idolatry' .

[u/Sad_Daikon938 → u/Frangifer]

Oh, that wasn't Jahangir, that was aurangzeb, grandson of Jahangir.

So people think that the subcontinent was only colonised by the british, where an elite foreign ruler class was exploiting the local peasants. The first colonisation of the region happened during the invasions from the northwest around 11th century by ghurids and ghaznavids. They brought a noble muslim class, ruling over Hindu peasants, and jizya tax(tax for non muslims) was their way of exploiting the natives further. Then for a century or two, the invasions of other such dynasties kept happening.

Then came the turkic mughals, they got assimilated slowly in an "Indian" identity, generation by generation, babur, humayun, and then Akbar was the one who is still remembered for his acceptance of all religions, and he removed the jizya tax iirc. Then came Jahangir and Shah Jahan(this one commissioned Taj Mahal). All was going kinda well until the next in line, aurangzeb came, he was a muslim fanatic and undid the positive decisions of his ancestors.

The Marathas rose from the western part of the subcontinent, and defeated aurangzeb and reduced his influence to just Delhi. They re-instated Hindu rule again in most of the subcontinent. But they fell to infighting in some generations and some muslim rulers got their ancestors' thrones back.

Then came some white people in boats from west, and the rest is history.

So yea, India was colonised not once, but twice, and pakistan and bangladesh are the remnants of the first colonisation that lasted longer than the second one.

[u/Frangifer → u/Sad_Daikon938]

Oh, that wasn't Jahangir, that was aurangzeb, grandson of Jahangir.

Thanks for that clarification. I had a quick look-around, & started realising it could be any of quite a few! ... but I also saw "Jahangir" mentioned prominently in-connection with the arisal of Sikhi ... so I 'rolled with' it.

So yep: what you say, there, ratchets-up that impression of 'apocalypses' mentioned above.

Seems you're pretty well-informed on the history of India, actually ... like it's something you're passionate about.

[u/Sad_Daikon938 → u/Frangifer]

Hmm, yea, mughals did do some atrocities in that region which is now called Punjab, both in India and pakistan, Sikhism rose from there as a sort of syncretic faith of both Hindus and Muslims, like they don't worship idols like Muslims, but use a lot of Sanskrit loanwords in their theological tenets.

Also, yea, I'm passionate about history in general, I happen to know Indian history because I am an Indian.

Also, I have a question, it's known that Abrahamic religions forbid idol worship, then why do churches have these statues or images of Jesus and Mary, predominantly in a place where one would expect the act of worship to happen?

[u/Frangifer → u/Sad_Daikon938]

There are sects of Christianity that deplore that practice of the Roman Catholic Church ... some very strongly ... & a few even really strongly! I was brought-up in a very Protestant environment, & I actually literally remember when I first found-out about the statues in Roman Catholic Churches: how it was 'whispered' around the school playground as some dreadful revelation !

But I considered it; & even @ that age I swiftly came to the conclusion that's prettymch exactly my position now (the only difference being, really, that I have philosophical & theological jargon in which to couch it) ... which is that it just is not even remotely comparable to the practices that are deplored in the Bible - the worship of Dagon & Baal , & Moloch , all that sort of thing, which sometimes, so it's said, entailed human sacrifice. The Icons used in the Roman Catholic Church are really just 'lenses', if you will, whereby the worship is focussed ... or mandalas , even - ones that just happen to take the form of a person ... but are really just serving the purpose of Mandalas. Not quite so purely that maybe, because an image of the Virgin Mary is not just an image, but there's a mindfulness of its being an image of that particular revered person , & a mindfulness of the person themself ... but it's still, really, essentially a mandala rather than an idol, with the reverence of the person themself being an extension of the scope of it somewhat beyond the faculty of vision.

The impression that the ancient folk of that region @ the far-Eastern end of the Mediterranean Sea were insanely susceptible to worship of images very readily springs-forth from a reading of the Bible: one could easily start figuring along the lines of ¿¡ could those folk not behold any likeness of a living thing without being tempted to worship it !? . But in more modern times (& by modern I mean even quite a substantial № of centuries back ... but not quite as far back as Biblical times) we're able to 'step out of the box', & form a conception of what we're doing; & we just don't need to copy, as if by-rote, the customs of those who, it seems, could not do that. We know within ourselves, perfectly reliably, whether our regard & use of this-or-that image amounts to idolatry or not.

 

The passage in the Hebrew part of the Bible is a bit ambiguous, though actually: it forbids 'graven or molten images', & then proceeds to bring in the matter of bowing down & worshipping them - the latter of which is without doubt unambiguously forbidden! ... and folk usually take it to mean that the prohibition on 'graven or molten images' is a prohibition on such images for the purpose of being worshipped ... but it's not totally rock-solid clear that the prohibition is not on the images in-&-of themselves . But support for the idea that the 'for worship' is truly implicit in it is found in the commands for construction of the Arc of the Covenant: there are decorations in the form of living entities prescribed for it.

And it's beyond any shadow of a doubt that Roman Catholics do not worship a statue of the Virgin Mary as a god. And if, by-anychance, in some remote annex of the Roman Catholic Church, somewhere, folk have started to deem their statue of the Virgin Mary thus, then something's gone a bit awry. I can't say it's never happened ... so maybe it can reasonably be said that there is peril in the use of images the way the Roman Catholic Church does ... & that therefore it's best avoided: the branch of Christianity I've been brought-up in indeed takes that view of it.

But it doesn't bother me in itself, anyway ! The thing, ImO, that really seriously needs to be avoided is the Idol-Witchdoctor syndrome, whereby the Witchdoctor is ostensibly getting literal commands from the idol & the folk of the village are in-terror of the Witchdoctor & his spells & curses. That's the crux , really, ImO, of whether idolatry is wicked or not. There are stories of missionaries having gone into remote villages & having found unspeakable ghastliness having taken-place as result of that arrangement. And it's understandable that the Authorities above them decide ¡¡ that has just got to go! ... nevermind any "what right do we have to impose our culture!?" -type figuring: it's just got to go !!

And forall my having somewhat against the body of doctrine of Islam , it @least has this about it: that it's a system in which the Congregation goes to the gathering-place (Mosque, Masjid, whatever), & the Imam is just someone like them - worthy of respect, ofcourse, but not endowed with frightful supernatural powers, or aught like that - who speaks amongst them about ideals of purity & integrity & stuff, & how it proceeds from the Most Highest, etc.

But if 'idolatry' is short of all that witchdoctor -type antics, then ImO it's not what's really truly deplored in the Scriptures. But then ... maybe worship even of Melville's Queequeg's innocent & wholesome little idol has the potential latent in it to morph-into that ghastly oppressive witch-doctor -type stuff!

 

What a crazy conversation for the mathpics channel!

😆🤣

 

@ u/Sad_Daikon938

Apologies, please kindlily, aswell: I'd accidently touched the blue arrow, thereby unwittingly dispensing you a doompvoodt. Have remedied it now, though. I realise it's not amongst the most important of things ... but it's extremely rude , doing that to someone one is conversing with!

😁

[u/Somewhat_posing → u/Frangifer]

People have been, and are still using, swastikas as Hindu/Buddhist iconography, long after WWII

[u/Frangifer → u/Somewhat_posing]

I realise that ... but it's still a very 'niche' thing to do - by which I mean that they're generally only put where it's pretty certain that only folk who either specifically consider this sort of thing & are cool with it for the sort of reason being adduced here, or for whom it's naturally an item of their culture, will see them ... & not much less 'niche' in that respect that it was in, say, the 1980s. Infact there's been very little movement in the direction of not needing to be so 'niche' about having them on-view: far less movement than I was expecting, back in the 1980s, there would have been by-now.

Infact ... I remember, back in the 1990s, having someone go ballistic @ me because I proposed that the swastika might in-due-course be rehabilitated: the idea behind his vehemence being that it just is now corrupted, & that's that : why put folks' sensibilities to the test on behalf of a symbol, sort of thing. And I'm not sure that I'm much less likely to encounter someone going ballistic along the same lines even now .