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depezblob · 5 months ago

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So what's up with dividing by zero anyways - a ramble on algebraic structures

Most everyone in the world (at least in theory) knows how to add, subtract, multiply, and divide numbers. You can always add two numbers, subtract two numbers, and multiply two numbers. But you must **never** divide by zero... or something along those lines. There's often a line of logic that leads to dividing by zero leading to "infinity," whatever infinity means, unless you're doing 0/0, whatever that means either. Clearly this is a problem! We can't have such inconsistencies in our fundamental operations! Why aren't our top mathematicians working on this?

So, that might be a bit of an exaggeration: division by zero isn't really a problem at all and is, for all intents and purposes fairly well understood, but to see why we'll have to take a crash course through algebra (the field of math, not the grade school version). Sorry for those of y'all who have seen fields and projective space before, not much to gain out of this one.

Part I: In the beginning, we had a Set.

As is true with most things in math, the only structure we start with is a set. A set isn't useful for much; all we can do with a single set is say what elements are and aren't in the set. Once you have more than one set, you start getting interesting things like unions or intersections or functions or Cartesian products, but none of those are _really_ that useful (or at least necessary) for understanding algebraic structures at the level we need, so a single set is what we start with and a single set there will be. The story then goes as follows: on the first day the lord said "Let there be an operation!" and it was so. If you want to be a bit of a nerd, a (binary) operation on a set A is formally a map * : A x A -> A, but for our purposes we just need to know that it matches the standard operations most people know (i.e. addition, subtraction, multiplication, but not division) in that for any two numbers a and b, we can do a * b and get another number. Of course, once again this is not very helpful on its own, and so we need to impose some more conditions on this operation for it to be useful for us. Not to worry though, these conditions are almost always ones you know well, if not by name, and come rather intuitively.

The first structure we'll discuss is that of a monoid: a set with an operation that is associative and has an identity. Associativity simply means that (a * b) * c = a * (b * c), and an identity simply means that we have some special element e such that a * e = e * a = a. For two simple examples and one nonexample, we have the natural numbers (with 0) under addition is a monoid: 0 + a *= *a *+ 0 = *a, and any two natural numbers add to another natural number; the integers under multiplication is a monoid: 1 * a = a * 1 = a, and any two integers multiply to another integer; and the integers under subtraction is not a monoid, since subtraction is not associative (a - (b - c) =/= (a-b) - c). In both of these examples, the operation is commutative: in other words, a * b = b * a for every a and b. There are plenty of examples of operations that are not commutative, matrix multiplication or function composition probably being the most famous, but for the structures we're going to be interested in later operations are almost always commutative, so we can just assume that from the start.

Of course, you might wonder where subtraction comes from, if it doesn't fit into a monoid structure (and in particular isn't associative). Not to worry! We can simply view subtraction as another type of addition, and our problems go away. In particular, we add the condition that for every a, we have an inverse element a⁻ ¹ (or -a if our operation is addition) such that a * a⁻ ¹ = a⁻ ¹ * a = e. For fans of universal algebra, just as a binary operation can be thought of as a function, the inverse can be thought of as a function i : A -> A that sends each element to its inverse. This forms a structure we know as a group. While none of the above examples form a group, one of them can be naturally extended to a group: if we simply add negative whole numbers to natural numbers, we get the group of integers over addition, where for any integer a, we have its inverse -a where a + -a = 0. In particular, the subtraction a - b is just a + -b = -b + a, where -b is the additive inverse of b. As we will soon see, division can also be thought of in a similar way, where a/b = a * /b = /b * a where /b is the multiplicative inverse of b. As a side note, the examples above are very specific types of monoids and groups which turn out to be quite far from the general ideas that monoids and groups are trying to encapsulate. Monoids show up often in computer science as they're a good model for describing how a list of commands affects a computer, and groups are better thought of as encapsulating symmetries of an object (think of the ways you can rotate and reflect a square or a cube).

Part II: So imagine if instead of one operation, we have... two...

If you've ever taken introductory algebra, you've probably never heard of monoids and only done groups. This is partially because monoids are much less mathematically interesting than groups are and partially because monoids are just not as useful when thinking about other things. For the purposes of this post, however, the logical steps from Set -> Monoid -> Group are surprisingly similar to the steps Group -> Ring -> Field, so I've chosen to include it regardless.

Just as we started from a set and added an operation to make a monoid, here we start from an additive group (i.e. a group where the operation is addition) and add another operation, namely multiplication, that acts on the elements of the group. Just like in the monoid, we will impose the condition that multiplication is associative and has an identity, namely 1, but we also impose the condition that multiplication meshes nicely with addition in what you probably know as the distributive properties. What we end up with is a ring, something like the integers, where you can add, subtract, and multiply, but not necessarily divide (for example, 2 doesn't have a multiplicative inverse in the integers, as a * 2 = 1 has no solutions). Similarly, when we add in multiplicative inverses to every nonzero element, we get a field, something like the rational numbers or the real numbers, where we can now divide by every nonzero number. In other words, a ring is an additive group with a multiplicative monoid, and a field is an additive group with a subset that is a multiplicative group (in particular the subset that is everything except zero). For those who want to be pedantic, multiplication in a ring doesn't have to be commutative, but addition is, and both addition and multiplication are commutative in a field. A full list of the conditions we impose on the operations of a monoid, group, ring, and field can be found here).

So why can't we have a multiplicative inverse to 0 in a field? As it turns out, this is because 0 * a = 0 for every a, so nothing times 0 gets you to 1. There is technically a structure you can have if 0 = 1, but it turns out there's only the one single element 0 in that structure and nothing interesting happens, so generally fields specifically don't allow 0 = 1. Then, what if instead we relaxed the condition that 0 * a = 0? Similarly, it turns out that this isn't one of the fundamental conditions on multiplication, but rather arises from the other properties (a simple proof is a * 0 = a * (0 + 0) = a * 0 + a * 0 implies 0 = a * 0 - a * 0 = a * 0 + a * 0 - a * 0 = a * 0). If we were to relax this condition, then we lose some of the other nice properties that we built up. This will be a recurring theme throughout the rest of this post, so be wary.

Part III. We can't have everything we want in life.

While all the structures so far have been purely algebraic and purely algebraically motivated, the simplest way to start dividing by zero is actually "geometric," with several different ways of constructing the same space. The construction we'll use is as follows: take any field, particularly the real numbers or the complex numbers. We can always take the cartesian product of a field K with itself to form what's called affine space K^2, which is the set of ordered pairs (a,b) for a, b in K. As a side note, the product of groups, rings, or fields has a natural definition of addition or whatever the underlying group operation is by doing it componentwise, i.e. (a,b) * (c,d) = (a * c, b * d), but our operations will not coincide with this, as you'll see soon. This affine space is a plane - in fact, when we do this to the real numbers, we get the Cartesian plane - within which we can construct lines, some of which we get by considering the set of points (x, y) satisfying the familiar equation y = mx + b for some 'slope' m and 'intercept' b. In particular, we want to characterize all the lines through the origin. This gives us all the lines of the form y = mx, as well as one additional line x = 0. This is the basic construction of what we call the projective line, a space characterizing all the lines through the origin of affine 2-space. The geometric picture of this space is actually a circle: the bottom point representing the number 0; the left and right halves representing negative and positive numbers, repsectively; and the top point representing the number "infinity."

There are a few ways of describing points on the projective line. The formal way of doing so is by using what are called homogenous coordinates. In other words, for any nonzero point (a,b) in affine space, it is surely true that we can find a line through the origin and (a,b). In particular, if a is not zero, then this line takes the form y = (b/a) x where the slope is b/a. Furthermore, any two points (a,b) and (c,d) can actually sit on the same line, in particular whenever c = ka and d = kb for some number k. Thus, we can define homogenous coordinates as the set of points [a : b] for a, b in our field where [a : b] = [ka : kb] by definition, and the point [0 : 0] is not allowed as it doesn't specify any particular line (after all, every line passes through the origin). As is alluded to above, however, this means that whenever a =/= 0, we can take k = 1/a to get [a : b] = [1 : b/a], in other words characterizing each line by its slope. Furthermore, whenever a = 0, we can take k = 1/b to get [0 : b] = [0 : 1]. In other words, the projective line is, as we informally stated above, equivalent to the set of slopes of lines through the origin plus one other point representing the vertical line, the point at "infinity." Since slopes are just numbers in a field, we can add, subtract, multiply, and divide them as we normally do with one exception: the slope of the lines containing [a : b] for any a =/= 0 is b/a, so clearly the line with infinite slope consisting of points [0 : b] implies that b/0 should be infinity. Voila! We can divide by zero now, right? Well... there are two loose ends to tie down. The first is what infinity actually means in this case, since it is among the most misunderstood concepts in mathematics. Normally, when people bandy about phrases such as "infinity isn't a number, just a concept" or "some infinities are different from others" they are usually wrong (but well meaning) and also talking about a different kind of infinity, the ones that arise from cardinalities. Everything in math depends on the context in which it lies, and infinity is no different. You may have heard of the cardinal infinity, the subject of Hilbert's Hotel, describing the size of sets and written primarily with aleph numbers. Similarly, you may also have heard of the ordinal infinity, describing the "place" in the number line greater than any natural number. Our infinity is neither of these: it is to some extent an infinity by name only, called such primarily to take advantage of the intuition behind dividing by zero. It's not "greater" than any other number (in fact, the normal ordering of an ordered fields such as the real numbers breaks down on the projective line), and this is a consequence of the fact that if you make increasingly negative and increasingly positive slopes you end up near the same place: a vertical line. In other words, "negative infinity" and "positive infinity" are the same infinity.

The second loose end is that defining our operations this way is actually somewhat algebraically unsound, at least with respect to the way we think about operations in groups, rings, and fields. As mentioned above, the operation of addition can be lifted to affine space as (a,b) + (c,d) = (a+c,b+d). However, this same operation can't really be used for homogenous coordinates, since [1, 0] = [2, 0] as they lie on the same line (the line with slope 0), but [1, 0] + [1, 1] = [2, 1] while [2, 0] + [1, 1] = [3, 1], and [2, 1] and [3, 1] are not the same line, as they have slopes 1/2 and 1/3, respectively. Dividing by zero isn't even needed to get weirdness here. Luckily, we can simply define new operations by taking inspiration from fractions: b/a + d/c = (bc + ad)/ac, so we can let [a : b] + [c : d] equal [ac : bc + ad] (remembering that homogenous coordinates do to some extent just represent the slope). Luckily, multiplication still works nicely, so we have [a : b] * [c : d] = [ac : bd]. Unluckily, with these definitions, we no longer get a field. In particular, we don't even have an additive group anymore: [a : b] + [0 : 1] = [0 : a] = [0 : 1], so anything plus infinity is still infinity. In other words, infinity doesn't have an additive inverse. Furthermore, despite ostensibly defining infinity as 1/0, the multiplicative inverse of 0, we have that [1 : 0] * [0 : 1] = [0 : 0], by our rules, which isn't defined. Thus, 0 still doesn't have a multiplicative inverse and 0/0 still doesn't exist. It seems like we still haven't really figured out how to divide by zero, after all this. (Once again, if you want to read up on the projective line, which is a special case of projective space, which is a special case of the Grassmannian, in more depth.)

Part IV: I would say wheels would solve all our problems, if not for the fact that they just make more problems.

At this point, to really divide by zero properly, we're going to need to bite the bullet and change what dividing really means. Just as we can think of subtraction as adding the additive inverse (i.e. a - b = a + -b where -b was a number), we can start thinking of division as just multiplying by... something, i.e. a/b = a * /b, where /b is something vaguely related to the multiplicative inverse. We can already start doing this in the projective line, where we can define /[a : b] = [b : a], and it works nicely as [a : b] * [b : a] = [ab : ab] = [1 : 1] whenever neither a nor b is zero. This lets us rigorize the statements 1/infinity = 0, infinity/0 = infinity, and 0/infinity = 0, but doesn't really help us do 0/0 or infinity/infinity. Furthermore, note that because 0/0 =/= 1, /[a : b] isn't really the multiplicative identity of [a : b], it's just the closest we can get.

Enter the wheel! If 0/0 is undefined, then we can simply... define it. It worked so nicely for adding in infinity, after all - the picture of the point we added for infinity is taking a line and curling it up into a circle, and I like circles! Surely adding another point for 0/0 would be able to provide a nice insight just as turning a line into the projective line did for us.

So here's how you make a wheel:

You take a circle.

You add a point in the middle.

Yeah that's it. The new point, usually denoted by ⊥, is specifically defined as 0/0, and really just doesn't do anything else. Just like for infinity, we still have that a + ⊥ = ⊥ and a * ⊥ = ⊥ for all a (including infinity and ⊥). It doesn't fit into an order, it doesn't fit in topologically, it is algebraically inert both with respect to addition and multiplication. It is the algebraic formalization of the structure that gives you NaN whenever you fuck up in a calculator and the one use of it both inside and outside mathematics is that it lets you be pedantic whenever your elementary school teacher says "you can't divide by zero" because you can go "yeah you can it's just ⊥ because i've been secretly embedding all my real numbers into a wheel this whole time" (supposing you can even pronounce that).

Part V: So what was the point of all this anyways

The wheel is charming to me because it is one of the structures in mathematics where you can tell someone just asked a question of "what if this was true," built some space where it was, and just started toying with it to see what happens. It's a very human and very beautiful thing to see someone go against conventional knowledge and ask "what breaks when you allow 0/0" even if conventional knowledge does tend to be right most of the time. In this sense, perhaps the uselessness of the wheel is the point, that even despite how little ⊥ does from a mathematical lens, some people still took the time to axiomatize this system, to find a list of conditions that were both consistent and sufficient to describe a wheel, and genuinely do actual work seeing how it fits in within the universe of algebraic structures that it stays in.

While a wheel may not be used for much (it might be describable in universal algebra while a field isn't, though I'm not too well versed in universal algebra so I'm not actually entirely sure), every other structure discussed above is genuinely well studied and applicable within many fields inside and outside of math. For more viewpoints on what the projective line (and in general the projective sphere) is used for, some keywords to help you on your way are compactification of a set if you care about the topological lens, the real projective line or the Riemann Sphere if you care more about the analysis side, or honestly the entirety of classical algebraic geometry if that's your thing.

Another structure that might be interesting to look at is the general case of common meadows, an algebraic structure (M, 0, 1, +, -, *, /) where the condition of / being involutive (i.e. /(/x) is not always x) is relaxed, unlike a wheel where it is always involutive. Note that these structures are called meadows because the base structure they worked on is a field (get it? not our best work I promise mathematicians are funnier than this). These structures are at the very least probably more interesting than wheels, though I haven't checked them out in any amount of detail either so who knows, perhaps there isn't much of substance there either.

#algebra #commutative algebra #mathematics #math #mathblr #wheel theory

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esthermika · 5 months ago

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doing matrix inversion by hand is super robotgirlcore

#mika's thought cabin

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aparticularbandit · 1 year ago

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it is important to me that you understand that:

junko has no luck (and when she does, it's bad luck)

nagito has fantastic luck, but it always comes at a cost - for every moment of good luck, there is also a moment of bad luck; the higher the good luck he wants, the greater the sacrifice of bad luck he needs first (see case 5)

izuru has luck with no cost, which makes his luck better than nagito's; he's the inverse of junko, but his luck is downloaded, fake, contrived (like the rest of his talent is)

this must be combined with:

junko can calculate and anticipate anything provided she has access to the right dataset (the better she knows someone/something, the better she can manipulate them/it - this extends to talent as well; the more data she has on how someone's talent functions, the better she can imitate it, which is why she can't imitate ryota's animation as completely in the way that she wants (different talents have different rates in terms of how much she needs to analyze them, though))

nagito can deduce but he cannot mimic; he dabbles in manipulation, but he is much more of a servant to someone else's goals, which means his analysis is always in service to someone else (kyoko is similar but not quite the same; kyoko can get to the point of deducing fast enough to prevent things, which is a lot more similar to junko's analysis, just used in a different way; kyoko's more straight forward than junko is, which is why deduce and not analyze)

izuru's analysis functions the same as junko's does, except that he doesn't need to analyze someone else's talent to mimic it; he has had talent downloaded straight into him like neo in the matrix; if he wants more talents than the ones he already has, he's gonna need them downloaded again, probably

as a result:

junko's weakness is luck and incomplete data sets; an incomplete (or wrong) data set leads to a miscalculation, and when something relies on luck and cannot be precalculated or predicted, then junko falls. she relies hard on her analysis, which gives her a blind spot. (junko, however, is aware of this. that's why she has both nagito and izuru on deck; in the hope calculation that their luck, correctly used, will allow her to predict things that happen by chance as well)

nagito's weakness is his reliance on his luck. he believes in his luck. his luck will take care of him. he can't control how it works, and he can't control how the bad luck will happen or who it will happen to, but he doesn't need to analyze something for his luck to get him out of it (and the smaller the chance that something will happen, the more likely it will happen to him; his luck works on an inverse to everyone else's, basically, so what's the point of predicting things on what is most likely to happen when, for him, it will probably be the other way around)

izuru could cover his incomplete data sets with his luck, but tends to rely so completely on his luck to cover his ass that he stops analyzing and leaves himself open there - he should be the strongest of the two, but really tends to leave him with more weaknesses until he figures out how to use the two in conjunction; something junko has figured out but cannot do because her luck is not her own

#musings #danganronpa #junko enoshima #nagito komaeda #izuru kamukura #this is how i write them in #dr1 end rewrite fic #but also this is how i write nagito's luck #it's an inverse of an average person's luck; and for every good luck there is a cost re: bad luck #he can control this to an extent in terms of he can make the chance of something /for a normal person/ smaller /and thus give it greater #luck for him/ (see russian roulette in case 4)#but there is ALWAYS a cost #izuru's only been around for a year; he's still learning how to all the things #hajime is slightly better at this sort of thing #but izuru runs so much on the vibes of how something should happen that he forgets that bad vibes are still a thing #(izuru can be /overwhelmed/)

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blueikeproductions · 8 months ago

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So I’m seeing some ask the question if we should allow Rodimus Prime to be leader again after years of focusing on Optimus Prime.

Im of two minds of it.

I think you can, but I also don’t trust other fans or Hasbro to let that happen.

Not easily.

So despite Hasbro waffling on it in modern times, the Autobots and Decepticons aren’t foreign to non Optimus and Megatron leaders.

But the problem of course….

… Goes back to the original movie, where Optimus died after a final battle with Megatron, and their replacements Rodimus and Galvatron. Hasbro insisted Optimus needed to die despite being warned against it, but funnily enough, the first draft didn’t feature a definitive new leader. Magnus is given the Matrix and becomes Ultra Magnus, but he dies later on, and thus the Matrix ultimately gets opened by DANIEL, unleashing its power to destroy Unicron. Hot Rod never inherits the power in this draft despite still being the main character.

You might be wondering why Daniel in particular. In this version of events, Cybertron was already consumed by Unicron as a sort of “Screw you” to Galvatron, with the Earth as Unicron’s final target.

This scene hits differently had it been Earth instead of Cybertron. So I think the idea was an Earthling, with the help of Hot Rod, ultimately saving the day.

By coincidence (I think, as we didn’t have this info yet), IDW mirrored this idea with Unicron succeeding in destroying Cybertron, with Earth as his final target.

Rodimus ultimately came into being as the movie script evolved, and Hasbro had him poised to be the new leader…

…But then the movie came out and kids freaked out at Optimus dying, as those that warned Hasbro about suspected. Rodimus had a decent run but complaints from unhappy children won out, with Optimus returning in the cartoon and getting new toys, starting with his Powermaster toy.

As such modern creatives haven’t exactly been eager to return to Rodimus, and usually when he DOES appear, it’s mostly to beat the skrud out of him. Because of the modern Primes end with Optimus plot, usually Rodimus isn’t considered a Prime either, but there are exceptions.

Animated originally intended for Rodimus to be Optimus’ arrogant rival, but Hasbro wasn’t comfortable with Rodimus being a colossal jerk, so that became Sentinel.

Animated still got a sort of pot shot in by having Rodimus be infected by Cosmic Rust though.

Rodimus was the Prime during Prime Wars, but shenanigans saw him relinquish the Matrix, get possessed by Unicron and end the series by being stuck in a coma. The next Prime became Optimal Optimus, which in story still feels like a “Screw you!” moment to me regarding Hot Rod.

Hot Rod appeared in the Bay films, but unlike most instances, he actually got treated pretty well. The unfortunate part is he spoke with a bad French accent as an inverse of Bumblebee’s muteness.

Energon Rodimus is an odd one.

In Japanese media, he retained the Convoy rank, so Grand Convoy and Rodimus Convoy were active at the same time and butted heads initially on what to do about Unicron. In American media, they took away his rank, so he was just Rodimus, so the impact of two acting Primes with conflicting opinions didn’t really get expressed well. He was still treated well, a legendary figure among young Autobots like Hot Shot and Ironhide, with Rodimus and Hot Shot having an arc that started out as a never meet your heroes plot but the two grew to become friends once Optimus and Rodimus began to better understand each other. The only drawback is once the Rodimus plot was done, he became a support character in the background mostly, the focus going back to Optimus and Galvatron’s relationship.

And so we’re not here all day on the matter, besides IDW’s Rodimus ending up as a depressed drunk serving under Thunderclash, and Last Bot Standing’s … everything, the biggest “Screw you!” wound up being Deviations, which asks what happens if Optimus survived the movie. Apparently, in this version, just about everyone hates Hot Rod except Optimus and Daniel and berates the guy just for breathing. Also most of the 3rd season cast die horribly, and so does Hot Rod, opening the Matrix, but dying in the explosion that took out Unicron and the Galvatron replacement, an upgraded Starscream Combiner called Megascream.

Deviations is what happens when you let a fan with a huge chip on their shoulder about the movie and S3 write for Transformers. On the plus side, Kranix got to live this time, so that’s nice.

Still Rodimus is sort of the Gohan: meant to take over, but the fans said no, so plans changed. Now we have some Dragon Ball fans begging for Gohan to become the lead again years after the fact, with what feels like a middle ground being his involvement in the Universe Tourney and his starring role in Drsgon Ball Super Hero. Ironically, last I peeked into Super, as a tie in to Super Hero, teen Goten and teen Trunks were the leads for a while, so that’s nice, but still kinda dodgy about Gohan.

Now it feels like we’ve hit that threshold for Rodimus, but honestly? I think that ship has sailed. Hasbro has entertained other leaders, with the Decepticons being the more prominent example, but they’re still reluctant on the Autobots.

RiD15 Bee, Heatwave, RBA Hot Shot and … kinda sorta Twitch were leaders of their respective teams, but they still ultimately were under Optimus in the command structure despite largely being free to do what they needed. I’d argue the only ones that worked on this scenario were Bee and Heatwave, as I think most have forgotten Hot Shot, and Mo was more the leader of the Terrans than Twitch was…

I think the bottom line is you can do Rodimus or a new leader character, but you have to actually give a skrud and put in the effort to make them likable like RiD15 Bee, Optimus Primal and TFA Optimus, vs Cyberverse’s… everything and the lack of commitment on it and EarthSpark’s part to certain ideas.

Twitch SHOULD have worked, but it was clear the interest was more on Megatron. Cyberverse had an interesting idea with Hot Rod and Soundwave as the leaders, but they couldn’t commit with Hot Rod, going back to Optimus, but Hot Rod did become part of the new Council. I feel like part of that was Hasbro interfering and saying no, but Cyberverse also had a very bad habit of butterfly chasing, so who’s to say? Interestingly, Cyberverse is among some shows that states the war only ends when Megatron dies, and implies the War only waged on because of Optimus and Megatron’s inability to reconcile (but also Megatron’s ego). CV had an opportunity to go all in on Hot Rod as the Autobot Leader, but wouldn’t/couldn’t take it, which is too bad because Hot Rod and Soundwave had rival chemistry. Sadly we’ll never know how that could’ve gone elsewhere.

The IDW GoBots comic would lean in a similar direction.

In this universe, the GoBots were created by humans on Earth, but Cy-Kill was the first to rebel and start a war. This war would see the remnants of humanity devolved into tribal primitives and misshapen mutants akin to Planet of the Apes, with the Earth carefully cradled in a metallic shell known as the new home planet of the GoBots: GoBotron. The war between Leader-1 and Cy-Kill raged for centuries, with both sides and the GoBot Elders getting pretty fed up with their leaders’ petty feud. Once the final battle occurred, and both leaders died, the rest of the GoBots basically went “FINALLY!” and began to start a new, first ejecting Earth from Gobotron, allowing humanity to be reborn by the guidance of the Rock Lords and remaining Earth bound GoBots. The big twist being GoBotron would later evolve into Cybertron, with the GoBots’ new leader, Road Ranger, who would turn out to be Optimus Prime’s ancestor.

The remains of Leader-1 and Cy-Kill would be combined into a new GoBot, taking the best from both.

Their descendant was Starscream…

Anyway the point being, is both leaders ultimately died for profess to be made in some form. Hasbro isn’t going to unclench Optimus any time soon it would appear, making the matter of a true successor to Optimus in the modern era debatable. Especially in the context of TFOne, you can’t really … do Rodimus if Rodimus doesn’t have the connection to Galvatron, formally D-16/Megatron. I think there’s story potential in Megatron/Galvatron being distraught he killed his old friend and rival, that’s been toyed with before in Armada, where Megatron became depressed after Optimus died taking a shot from the Hydra Cannon. Megatron wasn’t particularly that interested in Hot Shot who was the stand in leader at the time, only finding true joy when Optimus miraculously revived thanks to the Mini-Cons. Even in Energon, when Galvatron is at his most sadistic and unstable state, he finds a sort of twisted comfort in fighting Optimus and still acts disgusted at not being able to fight his old rival due to Unicron’s essence interfering.

So it’s… tricky. It can be done, but the long-standing rivalry of Optimus and Megatron still stands. Galvatron appearing in CyberWorld is at least a step in the right direction, as was the rivalry between Bumblebee and Steeljaw, we just need creatives to commit to it, and if they use Rodimus, not fall back on old habits in treating Rodimus as the butt monkey.

I truly think we are getting to a stage where we need to earnestly try new Autobot and Decepticon leaders once again, especially since Hasbro/creatives still seems content to bench Megatron or have him be a repentant, sometimes sassy, Grunkle Vegeta to other characters.

Since I don’t really trust most to use Rodimus well, I stand by going back to a Bee and Motormaster/Steeljaw’s style rivalry would be the best, and then go from there in a post Optimus/Megatron situation.

#blueike productions #blueike #transformers #maccadam #g1 transformers #gobots #unicron trilogy #transformers animated #transformers movie #transformers earthspark

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kaddyssammlung · 3 months ago

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“dark days for your Solstice”

I like this one because it's an opposite. Our course you have winter Solstice versus summer solstice and when it's winter solstice then it is the shortest day and the longest night. But even winter Solstice is celebrated and gets called “return of the light”.

I don't remember the amount of time...36 hours after solstice when you body does not know if the light will return or not, its ready to die, more or less. But not really sure...I saw a video once where someone explained why that is and what's going on but I don't want to look things up. It's what I remember.

I also remember winter solstice of 2012: for a second it looked like that was it. The end of this whole planet. But we are still here and so is earth so....

And also “Lucia” or “Lucifer” it means bringer of light. In some countries a bit north from here they celebrate Sankta Lucia.

“dancing through the depths of Hellfire”

Why would you want to do that?!

Fire...hmm....reminds me of a great energy work course that I did it was called “Activate your wings”. We threw everything that we owned (not in a literal sense) into a blue flame, the flame of the Phoenix, so that we can be reborn and arise from ashes.

I love energy work.

So...fire so that something new can emerge.

And also Hellfire....something inside of you, your trauma, your demons (the damn entities that you keep feeding) all these things can transform you.

I like that line btw. Omg! No, actually, I love it :)

“on the winds that started from within”

Another line that makes me very exited because I just feel it.

That need to change. That distant voice that I could not hear for many years because I had buried it underneath my unhealthy coping strategies. It wanted me to pay attention to it....and then I finally did and it changed everything.

Going deep within yourself is the key to your personal freedom.

“My blood beats so alive, might tear right through my skin”

The line that I like the most....so I already wrote about it :)

“So, tell me what you meant by “living past your half-life”

Of course I know what half-life means. I studied chemistry for a whole semester.....but also I learned it in school. But then again...idk. It speaks to me but I have nothing connected with it.

“in lockstep with the universe”

Another one that seems filled with opposites. Lockstep makes me think about the military and I connect something negative with it.

Living with the universe is something positive for me so.....idk XD

“you're well-versed in the afterlife”

Yes...because the inversion of the inverse matrix is gone. No longer “misaligned” in a sense.

We are all misaligned just like earth's axis.....when we no longer are, then earth no longer is...

“You know I'm sanctified by what's below”

But why not above?! Everything is upside down...?!

“no matter what you do, no matter where you go”

Yeah well...you can't escape yourself. I tried......

#sleep token #sleep token lyrics #song emergence

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talonabraxas · 1 year ago

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The Dream Sabbat

‘The true Sabbat is simultaneously a state of Dreaming-consciousness and an extradimensional locus where the convocation of the living and the dead occurs and the Great Return which leads to a new becoming is achieved. The celebrants of the Sabbat gather in the twilit forests and the mist-shrouded meadows of Elphame and through the averse formulae of the infinite return, deliberately ’go backwards’ to that which lies behind all phenomena and consciousness, the ineffable source of all creation glyphed in the Witch-Mysteries by the Cauldron and the Cavern.

This mystical self-reversion or initiatic regression to the root of All is synonymous with the Horned God’s law of Misrule. It provides the inner metaphysic of ritual reversal, symbolised by the Backwards Prayer, the Widdershins Dance, and the black tapers ceremonial inversions characteristic of the Sabbat-Rite. All these infer the way of initiatic return and self-reversal to the ground and matrix of primeval unity which is the true state of Sabbatic ecstasy.

…The Dream-Sabbat is the supreme rite of the Witches, a total actualisation of the Great Mystery - all restrictions and bonds are overcome there. The separations between god, human and beast dissolve in a polymorphous inferno of extasis, the secret rapture of inner Witchdom. Thus the Sabbat is a dream, a dream of such potency that the profane world seems pallid and unreal by comparison. To enter into this sacred world of paradaisal night-revels requires consummate agility of the Dream-Body and the employment of techniques to sidestep and diminish the hold of profane perceptual conditioning, enabling the leap or flight to the ’Other Side’ to be effected.’ --from Masks of Misrule by Nigel A. Jackson

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rbrooksdesign · 8 months ago

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"EDMT-#23," digital, Oct. 2024, Reginald Brooks

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dosesofcommonsense · 9 months ago

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Posts from TruthHammer and BioClandestine.

#BioClandestine

We are witnessing a fascinating post-debate shift.

The ABC “fact-checks” were not true, which has inversely drawn MORE attention to these topics, aka the Streisand Effect.

-Springfield

-Late term abortion

-Gun confiscation

Public discourse on X has thoroughly debunked ABC’s “fact-checks” and brought video receipts for every single Kamala lie. As time goes on, public perception is falling more in favor of Trump, thanks to the reach and influence of citizen journalists on X, and other social media, circulating the video evidence that disproves Kamala/ABC’s lies from the debate.

This not only proves Kamala and ABC lied, but it’s further confirmation that the MSM are in bed with the DNC, and once again proves Trump was right about the media being corrupt and out to get him. The more the media attack him with such blatant bias, the more it validates Trump’s entire narrative, and discredits the Dems for weaponizing the media against Trump. They just proved a “Deep State” exists, while the entire world was watching.

Initially, I did not think this debate would have much impact on public perception or change the calculus in November, but as I observe the public’s response on social media, I think this performance from ABC woke up more people than I projected.

Normally the moderators are more subtle with their bias, but ABC made it so obvious, that some of the remaining normies are starting to notice. Glitches in the matrix.

#TruthHammer #bioclandestine #abc news #media bias #deep state #msm is the enemy

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lynns-bonkle-blog · 10 months ago

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Bionicle mask ideas I came up with while on holiday (except the first one which I came up with beforehand but fleshed out while I was there)

Mask of Recording

Lets the user record whatever they are currently observing, which then gets stored in the mask and can be played back in the form of a projection.

If a being that can use masks puts on a Mask of Projection that has recordings currently stored on it, they can access those recordings.

The Great version can store up to five recordings, and any audiovisual modifications (e.g a Toa of Earth's natural night-vision; a Toa of Sonics' enhanced hearing; hallucinations/visions) are applied to the recording.

The Noble version can store a single recording, which is low-quality, has high amounts of rolling shutter, and lacks any audiovisual modifications.

Mask of Vision

An attempt at making the Vision Mask from VNOG work outside that game's mechanics.

Increases the resolution of the user's vision, letting them see distant objects more clearly.

The Noble version has half the effect of the Great version.

Mask of Chemistry

Lets the user combine substances to create new ones.

Has a built-in visor that can flip down to protect the user's eyes.

The Great version lets the user view every possible combination of any substances, whereas the Noble version only shows the outcome of the current ratio of whichever substances the user has access to.

Mask of Alternate Pathways

Highlights shortcuts, hidden passages etc. in the user's field of view.

Mask of Returning

Causes thrown objects to return to the user.

The Great version makes the objects automatically target whichever of the user's hands are currently free, whereas the Noble version just targets the user as a whole.

Mask of Excavation

Allows the wearer to pull up any underground object, provided it weighs less than twice what they could carry with their hands.

The Noble version only works with things that the wearer could carry normally.

Mask of Acceleration

If the user is piloting a vehicle, they can make it reach its top speed much faster.

The Noble version takes longer but otherwise does the same thing.

Mask of Distraction

The user can create small, noisy bursts of coloured light on surfaces, designed to distract other beings.

The Noble version can only create the visual or audio component, but not both.

Mask of Hallucination

The user can forcibly alter a target's audiovisual perception of reality, based around a"keyword" that the user thinks of when as activate it, along with the feelings that the target associates with that keyword (e.g if the mask was used on a Matoran with the keyword "Karzahni", they would see the world as resembling Karzahni (the island), their friends as post-modification Karzahnite Matoran, and their Turaga as somewhat resembling Karzahni (the being); inversely, if the keyword was "Artakha", they'd see the world as resembling Artakha (the island), their friends as ornately-armoured Artakhan workers, and their Turaga as somewhat resembling Artakha (the being).

Despite the intensity of the hallucinations, they can't physically affect the target, and the feel of an object wouldn't match up with how the target perceives it.

Considered immoral.

Mask of Many Souls

Stores the minds of other beings, who can then take control of the wearer's body if the wearer is willing to let them.

The Great version can store 48, while the Noble version can only store 12.

Mask of Hacking

Allows the user to hack into machinery and fully-inorganic beings by projecting their mind into the target's code-matrix, allowing them to physically interact with the code.

Lacks eyeholes, with the wearer's eyes being covered by a boxy "visor" that prevents them from being distracted while hacking (as they can still see).

The Great and Noble versions differ only in the amount of time that the hacking process lasts, with the Great version allowing the user to access the code for over an hour while the Noble version lasts about half of that.

Mask of Perspective

Lets the user adjust their field of view. That's it that's all it does.

#bionicle #long post

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