When a geometric interpretation of imaginary numbers was at last proposed, long after they were invented, it was as though accomplished by central committee. The upshot was easily enough understood but also simplistic. In broad brushstroke here is what seems to have gone down.
The 3 dimensions of Descartes’ coordinate system-a number already deficient from the perspective of mandalic geometry-were reduced to just one. Of the real number axes then only the x-axis remained. This meant from the get-go that any geometric figure that ensued could encompass only linearity in terms of real numbers and dimensions. It was applicable only to a line segment, so the complex plane that resulted could describe just one real dimension and one imaginary dimension. It consecrated the number line in a single dimension, to exclusion of its proper habitation in two others besides. Strike one for imaginary numbers.[1]
With that as background let’s look now at the rotations described by this geometric interpretation of imaginary numbers in the context of the complex plane.
i in the complex or cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis By Loadmaster (David R. Tribble) (Own work) [CC BY-SA 3.0orGFDL],via Wikimedia Commons
The number 1 is the multiplicative identity element for real numbers and the number -1 is the reflection inversion element for real numbers. Put another way, the number one times any number equals that number; the number -1 times any number is a negative of that number or the inverse number through a reference point, usually taken as zero. Multiplying by 1 then leaves 1, -1, i and -i all unchanged. Multiplying by -1 changes -1 to 1, 1 to -1, i to -i, and -i to i. In terms of rotations in the complex plane, these changes all involve a rotation through 180 degrees. Multiplication of the number 1 by i changes it to i; i by i changes it to -1; -1 by i to -i; and -i by i to 1. These changes all involve rotations through 90 degrees. And finally, multiplication of 1 by -i changes it to -i; -i by -i changes it to -1; -1 by -i to i; and i by -i to 1: all changes involving rotations through -90 degrees.
The figure below shows another way to interpret these rotations that amounts to the same tbing: i1 = i; i2 =-1; i3 = -i; i4 = 1. Click to enlarge.
Four numbers on the real line multiplied by integer powers of the imaginary unit, which corresponds to rotations by multiples of the right angle. By Keφr [CC0],via Wikimedia Commons
I think a committee of some sort must have come up with this resplendent plan. For certain it was an Academy of Mathematics and Sciences that endorsed and enthroned it. All bow to central authority.
I had planned to include a comparison of imaginary numbers and probable numbers in this post as well but because that is a long discussion itself, it will have to wait till the next post. I might add it should prove well worth the wait.
Image: A drawing of the first four dimensions. On the left is zero dimensions (a point) and on the right is four dimensions (A tesseract). There is an axis and labels on the right and which level of dimensions it is on the bottom. The arrows alongside the shapes indicate the direction of extrusion. By NerdBoy1392 (Own work) [CC BY-SA 3.0orGFDL],via Wikimedia Commons
Notes
[1] Mathematician William Rowan Hamilton subsequently addressed this deficiency in 1843 with his quaternions, a number system that extends the complex numbers to three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or, in other words, as the quotient of two vectors. This complicated matters even more by introducing a non-commutative multiplication operation to the system, though to be fair the quaternion coordinate system has found some useful applications mainly for calculations involving 3-dimensional rotations, as in 3-dimensional computer graphics,computer vision, and crystallographic texture analysis. Still it becomes problematic when theoretical physics attempts use of quaternions in calculations pertaining to atomic and subatomic spaces where rotations do not actually take place. The conclusion to be drawn here is that quaternions can be usefully, if somewhat clumsily, applied to 3-dimensional macro-spaces but are inapproriate for accurate description of higher dimensional spaces. What is here unfortunate and misleading is that quaternions apparently do describe outcomes of events in the quantum realm to some partial degree, if not the mechanisms of the events themselves. Physicists would not long tolerate them were that not so.
[ADDENDUM (24 APRIL, 2016) Since writing this I’ve learned that quaternions are not currently used in quantum physics nor were they ever, to any great degree, in the past.]
In other words, sometimes the right answer can be reached by a wrong method. In the case under discussion here, we should note that it is possible for a rotation to mimic inversion (reflection through a point). A 90° rotation in two dimensions can mimic a single inversion in a single plane through an edge of a square, and a 180° rotation in two dimensions can mimic a single inversion through a diagonal of a square or two successive inversions through two perpendicular edges of a square. A 180° rotation in three dimensions can mimic three inversions through three mutually perpendicular edges of a square; a combination of one inversion through a diagonal of a square and another through an edge perpendicular to the plane of the first inversion; or a single inversion through a diagonal of the cube. Subatomic paricles exist as discrete or quantized entities and would follow such methods of transformation rather than rotations through a continuous space. Of course, transformations involving a diagonal would require more transformative energy than one involving a single edge.
Such patterns of relationship and transformation could no doubt be described in terms of quantum states and quantum numbers without too much difficulty by a knowledgeable theoretical physicist. Surely doing so could be no more difficult than using quaternions, which may give a correct answer while also misleading and limiting knowledge of the the true workings of the quantum realm by using an incorrect mechanism, one non-commutative to boot. Nature doesn’t approve of hat tricks like that.
Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)
Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where n = x + 1 - p. :)
We have already looked briefly at three of the more important Boolean operators or logic gates: AND, OR,andXOR.NOT just toggles any two Boolean truth values (true/false; on/off; yes/no). Here we introduce two new logic gates which do not occur in Boolean algebra. Both play an important role in mandalic geometry though.
We’ll refer to the first of these new operators or logic gates as INV standing for inversionorinvert. This is similar to Boole’s NOT except that it produces toggling betweeen yang/+ and yin/- instead of 1 and 0. Because it is based on binary arithmetic, Boole’s NOT has been thought of as referring to inversion also (as in ONorOFF). Although both ANDandINV act as toggling logic gates they have very different results in the greater scheme of things, since nature has created a prepotent disparity between a -/+ toggle and a 0/1 toggle in basic parameters of geometry, spacetime, and being itself. This makes Boole’s AND just a statement of logical opposition, notinversion.
Recognition of this important difference is built into mandalic geometry structurally and functionally, as it is also into Cartesian coordinate dynamics and the logic of the I Ching, but lacking in Boole’s symbolic logic. This is necessarily so, as there is no true negative domain in Boolean algebra. The OFF state of electronics and computers, though it may sometimes be thought of in terms of a negative state, is in fact not. It relates to the Western zero (0), not the minus one of the number line. Where Boolean algebra speaks of NOT 1 it refers specifically to zero and only to zero. When mandalic geometry asserts INV 1 it refers specifically to -1 and only to -1 . The inversion of yang then is yin and the inversion of yinisyang.[1] In the I Ching, Taoist thought, and mandalic geometry the two are not opposites but complements and, as such, interdependent.
The second added logic gate that will be introduced now is the REV operator standing for reversionorrevert. This operator produces no change in what it acts upon. It is the multiplicative identity element (also called the neutral elementorunit element), as INV is the inverse element. In ordinary algebra the inverse element is -1, while the identity element is 1. In mandalic geometry and the I Ching the counterparts are yinandyang, respectively. If Boolean algebra lacks a dedicated identity operator, it nonetheless has its Laws of Identity which accomplish much the same in a different way:
A = A
NOT A = NOT A
Again, Boolean algebra has no true correlate to the INV operator. There can be no sign inversion formulation as it lacks negatives entirely. Although Boolean algebra may have served analog and digital electronics and digital computers quite well for decades now, it is incapable of doing the same for any quantum logic applications in the future, if only because it lacks a negative domain.[2] It offers up bits readily but qubits only with extreme difficulty and those it does are like tears shed by crocodiles while feeding.
[1] Leibniz’s binary number system, on which Boole based his logic, escapes this criticism, as Leibniz uses 0 and 1 simply as notational symbols in a modular arithmetic and not as contrasting functional elements in an algebraic context of either the Boolean or ordinary kind.
In the field of computers and electronics, Boolean refers to a data type that has two possible values representing true and false. It is generally used in context to a deductive logical system known as Boolean Algebra. Binary in mathematics and computers, refers to a base 2 numerical notation. It consists of two values 0 and 1. The digits are combined using a place value structure to generate equivalent numerical values. Thus, both are based on the same underlying concept but used in context to different systems. [Source]
[2] Moreover, I expect physics will soon enough discover that what it now calls antimatter is in some sense and to some degree a necessary constituent of ordinary matter. I can already hear the loudly objecting voices declaring matter and antimatter in contact necessarily annihilate one another, but that need not invalidate the thesis just proposed. My supposition revolves around the meaning of “contact” at Planck scale and the light speed velocity at which subatomic particles are born, interact and decay only to be revived again in an eternal dance of creation and re-creation. Material particles exist in some kind of structural and functional homeostasis, not all that unlike the anabolic and catabolic mechanisms that by means of negative feedback maintain all entities of the biological persuasion in the steady state we understand as life. Physics has yet to get a full grip on this aspect of reality, though moving ever closer with introduction of quarks and gluons to its menagerie of performing particles.
Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)
Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where n = x + 1 - p. :)
I’ve participated in animator’s MAP, I chose Mulholland Dr. to animate, because I love this movie! Music: Foster The People - Pumped Up Kicks This is not commercial project.
I’ve tried QIWI wallet for commissioner’s payments, but it didn’t work. I tried boosty and god bless it work! I am happy to say I finally come back! The first commission after a long rest. Actually, it wasn’t rest. It was a hard work as an animation clean up superviser, but… nevermind!
I’m trying to find the possabilities to take commissions, and found QIWI wallet.
1.I’ll send my QIWI in personal chat as soon the commission will be accepted. 2.Preferred currency - USD 3.Prepayment required 4.Send me a note if you want to commission me
1 slot - 1 commission (CLOSED) ANIMATION pencil test with inbetweening 4 seconds 150 $ example:
(Technically called Nikhilam Navatashcaramam Dashatah) This is a quick and simple way to multiply any two numbers. It’s easiest when the numbers are both close to a power of ten, but it will always work. The first step is to chose a power of ten that the numbers are closest to. In my example I will find the product of 14 and 12. Since 12 and 14 are close to 10 I will chose 10. 14 is 4 more than 10, and 12 is 2 more than 10, so I will write these numbers off to the side, as shown.
+4 times +2 is 8 so I write this number on the right. Then I cross add the 14 and the 2 or I add 12 and 4 to get 16. I write this number to the left, and put these two numbers together to get the right answer 168. (Although I say “put these numbers together” what is actually going on is that 16 is being multiplied by 10 then 8 is added. Knowing this will be helpful when the number on the right is larger than the chosen power of ten.)
Here’s an example with larger numbers. Since they are closer to 100, 100 is used instead of 10. This time the numbers are less than the chosen power of ten, but the same method can be used. Multiply -8 by -11 to get 88 (write that on the right), and add 89 to -8, or 92 to -11 to get 81 (write that on the left). 81 is then multiplied by 100 (since that is the power of ten we chose) and 88 is added. Hence the correct answer to 92x89 is 8188. This is a neat trick, but why does it work? consider the following algebra:
(x+a)(x+b)=c x^2+ x*a+x*b+a*b=c x(x+a+b)+a*b=c
Say x is the power of ten we chose. Then a and b are the the two numbers that represent how far our factors are from the chosen power of ten.
Based upon the subs, it seems pretty clear that the rings that Gabriel and Tsurugi will be selling are one in the same with the prototype Tomoe carried willingly into battle.
She clearly knew where the Miraculous was and was not, so it was far easier for her not to entirely lie to the heroes.
I bet you these designer rings will be super pricey and most of our…dark holders, for lack of a better term will be either very wealthy like Chloe or the beneficiary of someone with wealth like Lila.
Considering we also know that Luka has been said to be a tragic figure, I would not be at all surprised if one of the rings found it’s way onto his finger.
Oooooh that’s a good point about the pricy thing. I was imagining that the rings would be some mass manufactured device for mass control of the population but your idea also sounds likely.
Then again, if those rings are only for Gabriel’s allies, why advertise them at all?
Episode Multiplication! That’s the mouse miraculous!! My clown brain is hoping for at least a crumb of Multimouse. Like at least a mention. An acknowledgement. AAAAAAAA I’mma shut up and watch it now
-Woahhhh wut? Felix Fathom? Not Graham de Vanily?
-Well damn, I have hundreds of tags to change now lol
-I guess Fathom is his father’s family name?
-Also why does Felix have an in universe wiki entry… unless he’s also famous like Adrien
-Ohai Su Han. Are you gonna say what I think you’re gonna say
-Lol yes, don’t let this mans speak
-BRO SU HAN CRYING?
-WAIT WAIT HE BARELY SHOWED UP IN SEASON 4 BECAUSE HE WAS WATCHING CAT VIDEOS????
-Do… do those videos include edits of Chat acting stupid
-Ohmygod Chat you said exactly what I was thinking
-Su Han gonna walk all the way to Tibet lol
-Alright now go after Felix go go go go
-I hope Felix causes more problems on purpose~
-Ooooh Felix loreeeeeee
-Half American huh
-Oh boy, Felix has had quite the busy life
-Just like Adrien…
-What is it with rich parents making their kids do absolutely everything
-Hello Amelie! How you doing
-Is Felix just gonna dip out
-Go into hiding oop
-OH NO MARINETTE IS DOING THE “it’s so quiet in here” THING
-Rare shot of Gabriel actually doing his job for his company
-Wait is that an outfit for himself?
-Oh wonderful… only when Gabriel thinks he is very close to victory and rewriting the universe does he let Adrien have his way
-After all, once the universe is recreated, nothing Adrien did in THIS universe would matter anymore right?
-Gabriel rebounded VERY quickly after Nathalie chewed him out
-Or maybe he’s dissociating (if that’s the right word for it)
-And Marinette is spiraling again
-Seems to be the thing these past 2 seasons
-Well, she’s always been one for spiraling
-But especially now oof
-THIS IS NOT A DRILL ADRIEN REALLY IS GUSHING ABOUT MARINETTE TO MARINETTE
-Ohai Nathalie
-I was wondering how the deal of Adrien technically going “missing” would be settled and it’s just… not?
-Adrien comes right home and nobody questions how he got there lol
-I had a feeling that would happen
-Oh wow! So straight up everyone knows that Alix is a time traveler now
-No funny business, just telling it as it is
-Ohai Tomoe. Long time no see!
-Are you here to explain why the Tsurugi family logo is in the op of the show now
-ADRIEN GO IN FOR THE SMOOCH
-MARINETTE WITH THAT FRAME PERFECT DODGE
-wait wut
-aww
-Wait a sec Mylene has a necklace that looks like a kwagatama
-Did she get one from Mullo??
-Or is it just a coincidence that she has a necklace in the shape of a kwagatama… magatama… thing
-Are they gonna 3d print a new Adrien lol
-Gabriel can’t just make a new one so easily anymore can he
-Ahhh, so Gabriel is doing the smart thing and NOT going gung ho and using every miraculous
-Because the kwamis nearly consumed his soul the last time he tried that
-He’s taking time to make PLANS now
-I forget the name of Tomoe’s akuma form
-IKARI GOZEN. There we go
-So the fan theory was correct! Combining an akuma with a miraculous for extra spicy flavor
-Tomoe sus…
-The rings we see in the op, the ring that Tomoe has… related?
-Ye I figured
-Gabriel has gone full Apple lol
-Btw isn’t it hilarious how every evil corporation is modeled after Apple
-Gabriel your hair
-So… a new Gabriel/Tsurugi brand technology ring thing
-That will probably have some sort of controlling effect over whoever wears it…
-Hello Felix
-Please ruin the day once more
-Unironically this show becomes more fun when you do lol
-And Duusu can just say Nathalie’s name, proof that kwamis aren’t barred from speaking the name of PREVIOUS holders
That episode really was something! And it goes against the theory that each episode would see a different miraculous recovered.
So… Adrien is just gonna be deepfaked now huh? The entirety of Gabriel’s advertising will be CGI.
That’s… funny.
Deeply curious as to what those rings are tho. Could they be the same as the rings shown in the op or are they not.
Because I can’t see Gabriel putting miraculous magic into a mass manufactured product. Surely that’s gonna backfire on him.
Unless… only he can activate it. He akumatizes someone wearing one of those rings and activates whatever miraculous ability he likes to boost the akuma.
Anyway another ~fun~ episode. The situation for the heroes has arguably only gotten worse! :D
