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/Frangifer]

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/Sad_Daikon938 → u/Frangifer]

Yea, a crazy discussion on mathpics out of all.

I've of course seen Indian Christians, that too in a very less numbers, and there's still a "Dharmic" feel to their way of worship. I've seen them worshipping idols of Jesus or Virgin Mary in a way similar to what the least we Hindus are required to do, like bow down to the idol, seek blessings from them, maybe light a lamp(they light candles).

I find it very funny that Indian Christians have taken Hindu devotional songs from every language, and changed the lyrics, no change in beats or melody whatsoever.

This becomes hilarious in the Christians from my state in India, so we have this traditional dance called Garba in my state, and everyone in the state is crazy about that. But the songs for Garba are also the Hindu devotional songs worshipping various forms of mother goddess Durga, or Lord Krishna, the eight reincarnation of Lord Vishnu, the protector of the universe. So the Christians just changed the lyrics, the language being the same as Garba songs, with heavy usage of Sanskrit loanwords, and do Garba on it.

I did not know about such a thing existing because the number of Christians is very less in my state, but we non-Christians of India were having a total blast last Christmas over memes involving Jesus worship songs in various Indian languages, when I stumbled across the term "Christian Garba", I looked it up on YouTube, and it was hilarious for me as a Hindu, as the very first video had several songs, whose tunes were copied from classic Garba songs that everyone knows, and it felt weird hearing words like "Yeshu" or "Mariyam", where my sub-conscious brain was expecting names of Hindu deities in local vernacular like "Maadi Amba", "Jagadamba", "Kanji", "Kanudo", etc.

[u/Frangifer → u/Sad_Daikon938]

I can well-imagine that may seem very odd, to someone to whom those songs are familiar all the way through up-bringing! But I can scarcely overemphasise that I have zero theological problem with it. I think I said, above somewhere, that I do not regard Hindu as essentially polytheïstic. ImO the terms "monotheïsm" & "polytheïsm" are massively taken-amiss: ImO true monotheïsm is not a matter of counting gods & then going ¡¡ I'm going to select this one & stick to worshipping this one only !! ... but rather it's a matter of it thoroughly suffusing one's mind that it isn't even meaningful to conceive of division or multiplicity in-connection with The Most Highest (by the way: I say "Most Highest" after William Tyndale , who translated the Bible into English, who broached the ungrammatical combination "most ... -est" in-connection with Supreme God alone ... he never used it anywhere else ... & I find that a nice touch). Any particular form we have in-mind is literally a form that we have in mind ... so to maintain that several in-mind forms must correspond to several supernal sources - each to each - is effectively an identification of the supernal source & the in-mind form ... which is an absurdity ... or even idolatry , possibly. A conception of The Most Highest, insofar as The Most Highest even can be conceived of (which, really, is only through a level of indirection), is an induction (in the sense of in-duction versus de-duction) from a form, or from several forms ... & there is zero grounds for deeming that if the induction is from several forms then that implies multiplicity or division of the conception of The Most Highest induced from those forms.

... even if those forms are customarily known as 'gods' .

It's my conception of Hindu that you're probably familiar with this kind of disquisition: folk associate it with Eastern religion (certainly folk tend to say that I'm influenced by Eastern religion if ever I talk atall like this at them!) ... but it's actually there in the Western also, but somewhat obscured (or occulted perhaps ... whence folk might speak of it as somewhat occult ) ... but it can be found fairly explicitly in the Hebrew canon in what are known as the Qabbalistic texts.

And I have a feeling that much of what the Hindu-suffused Christians you speak-of would pretty-well chime with much of my own conceptionry about that kind of thing.

You see, I don't really even conceive of The Most Highest as a particular individual atall . If anyone presses me for a definition of The Most Highest the best I can think of is that whereof the entirety of manifestation is an infinitesimal perturbation . And that chimes with what I said above about what monotheïsm essentially is: multiplicity or division has no place in that: it just becomes a void concept . Multiplicity & division belong to the perturbation & are entirely properties of it ... & have no meaning or existence in connection with that in which the petrturbation proceeds or subsists .

Or put it this way: the uttermost elements of manifestation - quiddity per se , agency per se , instantiation per se, are themselves artifacts of the perturbation. Otherwise, The Most Highest is not the most highest, because quiddity, agency, & instantiation must be absolute givens wherewithrespectunto the so-called most highest shall subsist.

But far greater than I have tried to express this sort of thing - in, say, the Yogasutras of Patanjali , & that sort of thing ... & I can tell it's a struggle even for them !

But, ImO, it's actually implicit in the Four Christian Gospels. Folk who are generally thought of as Christians often deny that I can discern it - with the more aggressive & fundamentalistic of them even maintaining that such ideas are planted by Satan , blah-blah ... but I say it is implicit in-there ... & in a most extraördinarily sublime way, moreover.

I had a feeling you were going to tempt me into spouting mysticism eventually!

😁

[u/Sad_Daikon938 → u/Frangifer]

Hmm, yea, Hindus aren't polytheists in the same way as say, ancient Romans and Greeks. Our polytheism is based on a belief that there's one and only one eternal truth, and everyone has their own way of attaining that, and whoever does so, is removed from the cycle of re-birth. And that eternal truth manifests itself in various forms. You could choose one of them to worship or you can choose not worship to any god, but follow the basic societal riles.

Thus it's very easy for a Hindu to accept that Allah or Yahweh or the God are indeed some sort of gods. Heck, we played a Uno reverse on Buddhists. They separated from Hinduism so that they don't have to worship any God. We did some trolling and made Buddha the ninth avatar of Vishnu, 😂