Visualizing quaternions (4d numbers) with stereographic projection

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<h4 id="title" class="style-scope ytd-metadata-row-renderer">Caption authors (Polish)</h4>

<div id="content" class="style-scope ytd-metadata-row-renderer"><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UCLY0sVSIlw5qERRhLSMk0tA">Fundacja Edukacja Przyszłości</a><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UCQYPNNQExri6AG5NmQjmvNg">KhanAcademyPoPolsku</a></div>

<h4 id="title" class="style-scope ytd-metadata-row-renderer">Caption authors (English)</h4>

<div id="content" class="style-scope ytd-metadata-row-renderer"><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UCDjmBvdOu23e4jyJ971keUQ">cycl0pentadiene</a><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UCM-qiC0SS-47KYLxW2Sx3vw">Rachmat Ridwan</a></div>

<h4 id="title" class="style-scope ytd-metadata-row-renderer">Caption authors (Spanish)</h4>

<div id="content" class="style-scope ytd-metadata-row-renderer"><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UC8tic89FnDFfRCNRHIEQY6w">MATECLIPS</a><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UCIgANqQeTg3llfnibt0gj8w">Javier Wagner</a></div>

<h4 id="title" class="style-scope ytd-metadata-row-renderer">Category</h4>

<div class="cue style-scope ytd-transcript-body-renderer active">What you're looking at right now is something called quaternion multiplication,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">or rather, you're looking at a certain representation of a specific motion happening on a four-dimensional sphere</div>

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<div class="cue style-scope ytd-transcript-body-renderer">being represented in our three-dimensional space</div>

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<div class="cue style-scope ytd-transcript-body-renderer">one which you'll understand by the end of this video.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Quaternions are an absolutely fascinating and often underappreciated number system from math.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Just as complex numbers are a two-dimensional extension of the real numbers,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">quaternions are a four-dimensional extension of complex numbers,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">but they're not just playful mathematical shenanigans.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">They have a surprisingly pragmatic utility for describing rotation in three dimensions and even for quantum mechanics.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The story of their discovery is also quite famous in math.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The Irish mathematician William Rowan Hamilton</div>

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<div class="cue style-scope ytd-transcript-body-renderer">spent much of his life seeking a three-dimensional number system analogous to the complex numbers,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and as the story goes, his son would ask him every morning</div>

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<div class="cue style-scope ytd-transcript-body-renderer">whether or not he had figured out how to divide triples and he would always say "no, not yet."</div>

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<div class="cue style-scope ytd-transcript-body-renderer">But on October 16th, 1843, while crossing the Broome Bridge in Dublin,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">he realized—with a supposed flash of insight—</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that what he needed was not to add a single dimension to the complex numbers,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">but to add two more imaginary dimensions:</div>

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<div class="cue style-scope ytd-transcript-body-renderer">three imaginary dimensions describing space</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and the real numbers sitting perpendicular to that in some kind of fourth dimension.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">He carved the crucial equation describing these three imaginary units into the bridge</div>

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<div class="cue style-scope ytd-transcript-body-renderer">which today bears a plaque in his honor showing that equation.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Now, you have to understand our modern notion of vectors with their dot product and the cross product and things like that</div>

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<div class="cue style-scope ytd-transcript-body-renderer">didn't really exist in Hamilton's time, at least not in a standardized form.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So after his discovery, he pushed hard for quaternions to be</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the primary language with which we teach students to describe three-dimensional space,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">even forming an official quaternion society to proselytize his discovery.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Now, unfortunately, this was balanced with mathematicians on the other side of the fence</div>

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<div class="cue style-scope ytd-transcript-body-renderer">who believed that the confusing notion of quaternion multiplication</div>

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<div class="cue style-scope ytd-transcript-body-renderer">was not necessary for describing three dimensions</div>

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<div class="cue style-scope ytd-transcript-body-renderer">resulting in some truly hilarious old-timey trash talk legitimately calling them evil.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It's even believed that the Mad Hatter scene from Alice in Wonderland</div>

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<div class="cue style-scope ytd-transcript-body-renderer">whose author you may know was an Oxford mathematician</div>

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<div class="cue style-scope ytd-transcript-body-renderer">was written in reference to quaternions: that the chaotic table placement changes were mocking their multiplication,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and that certain quotes were referencing their non-commutative nature.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Fast forward about a century and the computing industry gave</div>

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<div class="cue style-scope ytd-transcript-body-renderer">quaternions a resurgence among programmers who work with graphics and robotics and anything involving orientation in 3D space,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and this is because they give an elegant way to describe and to compute 3D rotations</div>

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<div class="cue style-scope ytd-transcript-body-renderer">which is computationally more efficient than other methods and which also avoids a lot of the numerical errors that arise in these other methods.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The 20th century also brought quaternions some more love from a completely different direction, quantum mechanics.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">You see, the special actions the quaternions describe in four dimensions</div>

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<div class="cue style-scope ytd-transcript-body-renderer">are actually quite relevant to the way that two-state systems like spin of an electron, or</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the polarization of a photon are described mathematically.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">What I'll show you here is a way to visualize quaternions in their full four-dimensional glory.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It would surprise me if this approach was fully original,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">but I can say that it's certainly not the standard way to teach quaternions, and that these specific</div>

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<div class="cue style-scope ytd-transcript-body-renderer">four-dimensional right-hand rule image that I'd like to build up to is something that I haven't really seen elsewhere.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Building up an understanding for this visual will take us meaningful time,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">but once you have it, there is a very natural and satisfying intuition for how to think about quaternion multiplication</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It won't be until the next video that I show you how exactly quaternions describe orientation in three dimensions,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">which is for some people the whole reason we care about it,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">but once we're able to go at it armed with the image of what they're doing to a 4d hypersphere,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">there's a pleasing understanding to be had for the otherwise opaque formulas characterizing this relationship.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The structure here will be to start by imagining teaching complex numbers to someone who only understands one dimension,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">then describing 3d rotations to someone who only understands two dimensions,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and ultimately to represent what quaternions are doing up in four dimensions,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">within the constraints of our 3d space.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Our first character is Linus the Linelander, whose mind can only grasp the one-dimensional geometry of lines and the algebra of real numbers.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">We're gonna try to describe complex members to Linus, and it's really important for you to</div>

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<div class="cue style-scope ytd-transcript-body-renderer">empathize with him as much as you can during this, because in a few minutes you're gonna be in his shoes</div>

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<div class="cue style-scope ytd-transcript-body-renderer">On the one hand, you could define complex numbers purely algebraically:</div>

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<div class="cue style-scope ytd-transcript-body-renderer">you say each one is expressed as some real number plus some other real number times i,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">where i is a newly invented constant whose defining property is that i × i = -1.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Then you say to Linus to multiply two complex numbers, you just use the distributive property—what many people learn in school as "FOIL"—</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and you apply this rule that i × i = -1 to simplify things down further. And that's fine!</div>

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<div class="cue style-scope ytd-transcript-body-renderer">That totally works, and the standard textbook way to introduce quaternions is analogous to this:</div>

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<div class="cue style-scope ytd-transcript-body-renderer">showing the algebraic rules and calling it done</div>

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<div class="cue style-scope ytd-transcript-body-renderer">But I think something is missing If we don't at least try to show Linus the geometry of complex numbers</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and what complex multiplication looks like, since the problems in math and physics</div>

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<div class="cue style-scope ytd-transcript-body-renderer">where complex numbers are shockingly useful often leverage this spatial intuition.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">You and I, who understand two dimensions, might think of it like this:</div>

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<div class="cue style-scope ytd-transcript-body-renderer">When you multiply two complex numbers, z times w,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">you can think of z as a sort of function acting on w, rotating and stretching it in some way.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">I like to think of this by broadening the view and asking "what does z do to the entire plane?"</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and you can think of that bird's-eye view action by imagining using one hand to fix the number 0 in place,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and using another hand to drag the point at 1 up to z,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">since anything times 0 is 0 and anything times 1 is itself.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And in two dimensions, there is one and only one stretching-rotating action on the plane that'll do this.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This is also how I'll have you thinking about quaternion multiplication later on,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">where the number on the left acts as a kind of function to the one on the right, and</div>

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<div class="cue style-scope ytd-transcript-body-renderer">we'll understand this function by seeing how it acts by transforming space,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">although instead of rotating 2d space, it does a sort of double rotation in 4d space.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">By the way, if you want to review thinking about complex numbers as a kind of action,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">a good warm-up for this video might be the one I did on e to the pi i explained with introductory group theory</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Now Linus the Linelander is pretty comfortable with the idea of stretching: that's what multiplication by real numbers looks like.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Maybe it's a little weird for him to think about stretching in multiple dimensions, but it's not fundamentally different.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The difficult thing to communicate to Linus is rotation:</div>

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<div class="cue style-scope ytd-transcript-body-renderer">specifically, focus on the unit circle of the complex plane—all the numbers a distance 1 from zero—</div>

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<div class="cue style-scope ytd-transcript-body-renderer">since multiplication by these numbers corresponds to pure rotation.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">How would you explain to Linus the look and the feel of multiplying by these numbers?</div>

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<div class="cue style-scope ytd-transcript-body-renderer">At first, that might seem impossible. I mean, rotation is just such an intrinsically two-dimensional idea.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">But on the other hand, rotation involves only one degree of freedom:</div>

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<div class="cue style-scope ytd-transcript-body-renderer">a single number, the angle, specifies a given rotation uniquely.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So in principle, it should be possible to associate the set of all rotations to the one-dimensional continuum that is Linus's world</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And there are many ways you could do this, but the one I'm going to show you is what's called a stereographic projection</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It's a special way to map a circle onto a line, or a sphere into a plane, or even a 4d hypersphere into 3d space.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">For every point on the unit circle, draw a line from -1 through that point</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and wherever it intersects the vertical line through the circle's center, that's where the point of the circle gets projected</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So for example, the point at 1 gets projected into the center of the line;</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the point i actually stays fixed in place, as does -i;</div>

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<div class="cue style-scope ytd-transcript-body-renderer">all of the points on that 90° arc between 1 and i</div>

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<div class="cue style-scope ytd-transcript-body-renderer">will get projected somewhere in the interval between where 1 landed and where i landed.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">As you continue farther around the circle on the arc between i and -1,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the projected points end up farther and farther away at an increasing rate.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Similarly, if you come around the other way towards -1,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the projected points end up farther and farther on the other end of the line.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This line of projected points is what we show to Linus,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">labeling a few key points like 1 and i and -1, all for reference.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Technically, the point at -1 has no projection under this map,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">since the tangent line to the circle at that point never crosses the vertical line,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">but what we say is that -1 ends up at the "point" at infinity.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This is a special point you imagine adding to the line where you would approach it</div>

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<div class="cue style-scope ytd-transcript-body-renderer">if you walk infinitely far along the line in either direction.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Now it's important to remember and to remind Linus that what he's seeing is only the complex numbers that are a distance 1 from the origin: the unit circle.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Linus doesn't see most numbers like 0 or 1 + i or -2 - i.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">But that's okay, because right now we just want to describe complex numbers z</div>

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<div class="cue style-scope ytd-transcript-body-renderer">where multiplying by z has the effect of a pure rotation, so he only needs to understand the unit circle.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">For example, when we take the number i and multiply it by any other complex number w,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the effect is to rotate by 90° counterclockwise.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">And when we apply this action to the circle being projected down to the line for Linus, what does he see?</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Well, it's a bit of a strange morphing action on the line, one which I want you to become familiar with for something we'll see later on.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">It's easiest to understand by following a few key reference points.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">i times 1 is i, so that means the number 1 should move up to i.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">i times i is -1, so the point at i slides off to infinity.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">i times -1 is equal to -i, so that point at infinity kind of comes back around from the bottom to the position one unit below the center</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and i times -1 is 1, so that point at -i slides up to 1.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Even though this is kind of a weird motion, it lets us communicate some important ideas to Linus.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">For example, multiplying by i four times, which corresponds to rotating by 90° four times in a row,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">gets us back to where we started: i to the fourth equals one.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Here, to get more of a feel for things</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">let me just show the circle rotated at various different angles, on both the left and the right half of the screen here,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and putting a hand on the point that started at the number 1 to help us and to help Linus keep track of the overall motion.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Next, let's introduce Felix the Flatlander, who only understands two-dimensional geometry.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Imagine trying to explain rotations of a sphere to Felix.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">In the spirit of transitioning from complex numbers to quaternions,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">let's extend the complex numbers with its horizontal axis of real numbers and its vertical axis of imaginary numbers</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">with a third axis, defined by some newly invented constant j</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">sitting one unit away from 0, perpendicular to the complex plane.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Instead of having this new axis in the z direction like you might expect, for a better analogy with how we'll visualize quaternions,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">we'll want to orient things so that the i and the j axes sit in the x and the y directions</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">with the real number line aligned along the z direction.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">So every point in 3d space is described as some real number,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">plus some real number times i, plus some real number times j.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">As it happens, it's not possible to define a notion of multiplication for a 3d number system like this</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that would satisfy the usual algebraic properties that make multiplication a useful construct.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Perhaps I'll outline why this is the case in a follow-on video, but staying focused on our current goal,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">think about describing 3d rotations in this coordinate system to Felix the Flatlander.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">The unit sphere consists of all those numbers which are a distance 1 from 0at the origin,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">meaning the sum of the squares of their coordinates is 1.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">We can't show all of 3d space to Felix, but what we can do is project this 2d surface to him</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and give him a feel for what reorientations of the sphere look like under that projection.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Analogous to what we did before, stereographic projection will associate almost every point on the unit sphere</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">with a unique point on the horizontal plane defined by the i and the j axes.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">For each point on the sphere, draw a line from -1 at the south pole through that point and see where it intersects the plane.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">So the point 1 at the north pole ends up at the center of the plane;</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">all of the points of the northern hemisphere get mapped somewhere inside the unit circle of the ij plane;</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and that unit circle which passes through i, j, -i, and -j actually stays fixed in place.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">And that's an important point to make note of:</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">even though most points and lines and patches that Felix the Flatlander sees are going to be warped projections of the real sphere,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">this unit circle is the one thing that he has which is an honest part of our unit sphere, unaltered by projection</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">All of the points in the southern hemisphere get projected outside that unit circle,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">each getting farther and farther away as you approach -1 at the south pole.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">And again, -1 has no projection under this mapping, but what we say is that it ends up at some point at infinity.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">That point at infinity is something such that no matter which direction you walk on the plane,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">as you go infinitely far out, you'll be approaching that point.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">It's analogous to how if you walk any direction away from the north pole, you're approaching the south pole.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Now let me just pull up a view of what Felix sees in two dimensions.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">As I rotate the sphere in various ways, the lines of latitude and longitude drawn on that sphere</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">get projected into various circles and lines in Felix's space.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">And the way I've done things up here, the checkerboard pattern on the surface of the sphere is accurately reflected in the projected view that you see with Felix,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and the pink dot represents where the point that started at the north pole ends up after the rotation,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and that yellow circle represents where the Equator ended up after the projection</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">The more you put yourself in Felix's shoes right now, the easier quaterniums will be in a moment.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">And as with Linus, it helps to focus on a few key reference objects, rather than trying to see the whole sphere.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">This circle, passing through 1, i, -1, and -i, gets mapped onto a line which Felix sees as the horizontal axis.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">It's important to remind Felix that what he sees is not the same thing as the i axis.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Remember, we're only projecting the numbers that have a distance 1 from the origin,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">so most points on the actual i axis, like 0 and 2i and 31 and et cetera, are completely invisible to Felix</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Similarly the circle that passes through 1, j -1, and -j gets projected onto what he sees as a vertical line.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">And in general, any line that Felix sees comes from some circle on the sphere that passes through -1.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">In some sense, a line is just a circle that passes through the point at infinity.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Now think about what Felix sees as we rotate the sphere.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">A 90° rotation about the j axis brings 1 to i, i to -1, -1 to -i, and -i to 1.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">So what Felix the Flatlander sees is an extension of the rotation that Linus the Linelander was seeing.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Notice also that this action rotates the ij unit circle to the position where the 1-j unit circle used to be</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">So what Felix sees is his yellow unit circle getting transformed into a vertical line,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">while that red vertical line gets transformed into the unit circle.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Of course, from our perspective, we know this is all just rigid motion.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">No actual stretching or more thing is taking place; all of that is just an artifact of the projection.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Similarly a rotation about the i axis involves moving 1 to j, j to -1, -1 to -j, and -j to 1.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">This rotation turns the ij unit circle into the 1-i unit circle,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">which, to Felix, looks like the unit circle getting transformed into a horizontal line.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">A rotation about the real axis is actually quite easy for Felix to understand</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">since the whole projection simply gets rotated about the origin,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">where the only point staying fixed in place are 1 at the origin and -1 off at infinity.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">In the same way that the complex numbers included the real numbers</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">with a single extra quote-unquote "imaginary dimension" represented by the unit i,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and that the not-actually-a-number-system thing we had in three dimensions included a second imaginary direction j,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the quaternions include the real numbers together with three separate imaginary dimensions,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">represented by the units i, j, and k.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Each of these three imaginary dimensions is perpendicular to the real number line,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and they're all perpendicular to each other somehow.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">So in the same way that complex numbers are represented as a pair of real numbers,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">each quaternion can be written using four real numbers, and it lives in four-dimensional space.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">You often think of this as being broken up into a real or "scalar" part, and then a 3d imaginary part.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">And Hamilton used a special word for quaternions that had no real part and just ijk components,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">a word which was previously somewhat foreign in the lingo of math and physics: "vector".</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">On the one hand, you could just define quaternion multiplication by giving the rules for how i, j, and k multiply together</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and saying that everything must distribute nicely.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">This is analogous to defining complex multiplication by saying that i times i is -1,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and then distributing and simplifying products.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">And indeed, this is how you would tell a computer to perform quaternion multiplication,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and the relative compactness of this operation compared to, say, matrix multiplication,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">is what's made quaternions so useful for graphics programming and many other things.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">There's also a rather elegant form of this multiplication rule written in terms of the dot product and the cross product,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and in some sense, quaternion multiplication subsumes both of these notions—at least, as they appear in three dimensions.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">But just as a deeper understanding for complex multiplication comes from understanding its geometry,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that multiplying by a complex number involves a combination of scaling and rotating,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">you and I are here for the four-dimensional geometry of quaternion multiplication.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">And just as the magnitude of a complex number—its distance from zero—</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">is the square root of the sum of the squares of its components,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that same operation gives you the magnitude of a quaternion.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">And multiplying quaternion, q1, by another, q2, has the effect of scaling q2 by the magnitude of q1</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">followed by a very special type of rotation in four dimensions.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">And those special 4d rotations, the heart of what we need to understand,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">correspond to the hypersphere of quaternions a distance 1 from the origin,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">both in the sense that the quaternions whose multiplying action is a pure rotation live on that hyper sphere,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and in the sense that we can understand this weird 4d action just by following points on the hypersphere,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">rather than trying to look at all of the points in the inconceivable stretch as a four-dimensional space.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Analogous to what we did for Linus and Felix, we stereographically project this hypersphere into 3d space</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">This label in the upper right is going to show a given unit quaternion,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and this little pink dot will show where that particular quaternion gets projected in our 3d space.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Just as before, we're projecting from the number -1,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">which sits on the real number line that is somehow perpendicular to all of our 3d space and beyond our perception.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Just as before, the number 1 ends up projected straight into the center of our space,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and in the same way that i and -i were fixed in place for Linus,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and that the ij unit circle was fixed in place for Felix,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">we get a whole sphere passing through i, j, and k on that unit hypersphere which stays in place under the projection.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">So what we see as a unit sphere in our 3d space</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">represents the only unaltered part of the hypersphere of quaternions getting projected down on to us.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">It's something analogous to the equator of a 3d sphere, and it represents all of the unit quaternions whose real part is 0.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">What Hamilton would have described as unit vectors.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">The unit quaternions with positive real parts between 0 and 1</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">end up somewhere inside this unit sphere closer to the number 1 in our 3d space,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">which should feel analogous to how the northern hemisphere got mapped inside the unit circle for Felix.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">On the other hand, all the unit quaternions with negative real part end up somewhere outside that unit sphere.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">The number -1 is sitting off at the point at infinity, which you can easily find by walking in any direction.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Keep in mind, even though we see the projection of some of these quaternions as being closer or farther from the origin of our 3d space,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">everything you're looking at represents a unit quaternion,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">so everything you're looking at really has the same magnitude: the same distance from the number 0.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">And that number 0 itself is nowhere to be found in this picture. Like all other non-unit quaternions, It's invisible to us.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">In the same way that, for Felix, the circle passing through 1, i, -1, and -i got projected into a line through the origin,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">when we see this line through the origin passing through i and -i, we should understand that it really represents a circle.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Likewise, up on the hypersphere invisible to us, there is a unit sphere passing through 1, i, j, -1, -i, and -j,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and that whole sphere gets projected into the plane that we see passing through 1, i, -i, j, -j, and -1 off at infinity:</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">what you and I might call the xy plane.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">In general, any plane that you see here really represents the projection of a sphere</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">somewhere up on the hypersphere which passes through the number -1.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Now the action of taking a unit quaternion and multiplying it by any other quaternion from the left</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">can be thought of in terms of two separate 2d rotations happening perpendicular to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and in sync with each other in a way that could only ever be possible in four dimensions.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">As a first example, let's look at multiplication by i.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">We already know what this does to the circle that passes through 1 and i, which we see as a line.</div>

</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">i goes to -1 off at infinity,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">-1 comes back around to -i, and -i goes to 1.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Remember, just like what Linus saw, all of this is the stereographic projection of a 90° rotation.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Now look at the circle passing through j and k,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">which is in a sense perpendicular to the circle passing through 1 and i.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Now, it might feel weird to talk about two circles being perpendicular to each other,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">especially when they have the same center, the same radius, and they don't touch each other at all,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">but nothing could be more natural in four dimensions.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">You can think of the action of i on this perpendicular circle as obeying a certain right-hand rule.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">If you'll excuse the intrusion of my ghostly green screen hand into our otherwise pristine platonic mathematical stage,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">you let that thumb of your right hand point from the number 1 to i, and you curl your fingers.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">The jk circle will rotate in the direction of that curl.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">How much? Well, by the same amount as the 1-i circle rotates, which is 90° in this case.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">This is what I meant by two rotations perpendicular to and in sync with each other.</div>

</div>

</div>

</div>

</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">This gives us a little table for what the number i does to the other quaternions.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">But I want this not to be something that you memorize,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">but something that you could close your eyes and you could really see.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Computationally, if you know what a quaternion does to the numbers 1, i, j, and k,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">you know what it does to any arbitrary quaternion, since multiplication distributes nicely.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">In the language of linear algebra, 1, i, j, and k form a basis of our four dimensional space,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">so knowing what our transformation does to them gives us the full information about what it does to all of space.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Geometrically, a four-dimensional creature would be able to look at those two perpendicular rotations that I just described</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and understand that they lock you into one and only one rigid motion for the hypersphere.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">We might lack the intuitions of such a hypothetical creature, but we can maybe try to get close.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Here's what the action of repeatedly multiplying by i looks like on our stereographic projection of the ijk sphere.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">It gets rotated into what we see as a plane,</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">then gets rotated further back to where it used to be (though the orientation is all reversed now),</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">then it gets rotated again into what we see as a plane, and after the fourth iteration it ends up right back where it started.</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">As another example, think of a quaternion like q = -(√2)/2 + (√2)/2 i</div>

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<div class="cue style-scope ytd-transcript-body-renderer">which, if we pull up a picture of a complex plane, is a 135° rotation away from 1 in the direction of i.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Under our projection, we see this along the line from 1 to i somewhere outside the unit sphere.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">If that sounds weird, just remember how Linus would have seen the same number.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The action of multiplying this q by all other quaternions</div>

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<div class="cue style-scope ytd-transcript-body-renderer">will look to us like dragging the point at 1 all the way to this projected version of q,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">while the jk circle gets rotated 135° according to our right-hand rule.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Multiplication by any other quaternion is completely similar.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">For example, let's see what it looks like for j to act on other quaternions by multiplication from the left.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The circle through 1 and j, which we see projected as a line through the origin, gets rotated 90°, dragging 1 up to j.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So j times 1 is 1, and j times j is -1.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The circle perpendicular to that one, passing through i and k,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">gets rotated 90° according to this right-hand rule, where you point your thumb from 1 to j.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So j times i is -k,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and j times k is i.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">In general, for any other unit quaternion you see somewhere in space,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">start by drawing the unit circle passing through 1, q, and -1,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">which we see in our projection as a line through the origin,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">then draw the circle perpendicular to that one on what we see as the unit sphere.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">You rotate the first circle so that 1 ends up where q was,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and rotate the perpendicular circle by the same amount. according to the right hand rule.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">One thing worth noticing here is that order of multiplication matters: it's not, as mathematicians would say, commutative.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">For example, i times j is k, which you might think of in terms of i acting on the quaternion j, rotating it up to k.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">But if you think of j as acting on i, j times i, it rotates i to -k.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">In fact, commutativity—this ability to swap the order of multiplication—is a way more special property than a lot of people realize,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and most groups of actions on some space don't have it. It's like how in solving a Rubik's cube, order matters a lot,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Or how rotating a cube about the z axis and then about the x axis</div>

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<div class="cue style-scope ytd-transcript-body-renderer">gives a different final state from rotating it about the x axis, then about the z axis.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And last, as one final but rather important point,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">so far I've shown you how to think about quaternions as acting by left multiplication,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">where when you read an expression like i times j,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">you think of i as a kind of function morphing all of space and j is just one of the points that it's acting on.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">But you can also think of them as a different sort of action:</div>

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<div class="cue style-scope ytd-transcript-body-renderer">by multiplying from the right, where in this expression, j would be acting on i.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">In that case, the rule for multiplication is very similar. It's still the case that 1 goes to j and j goes to -1, etc.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">But instead of applying the right-hand rule to the circle perpendicular to the 1-j circle, you would use your left hand.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So either way, i times j is equal to k,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">but you can either think about this with your right hand curling the number j to the number k as your thumb points from 1 to i,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">or as your left hand curling i to k as its thumb points from 1 to j.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Understanding this left hand rule for multiplication from the other side will be extremely useful</div>

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<div class="cue style-scope ytd-transcript-body-renderer">for understanding how unit quaternions describe rotation in three dimensions.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And so far, it's probably not clear how exactly quaternions do describe 3d rotation.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">I mean, if you consider one of these actions on the unit sphere passing through i, j, and k,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">it doesn't leave that sphere in place—it morphs it out of position.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So the way that this works is slightly more complicated than a single quaternion product.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">It involves a process called conjugation,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and I'll make a full follow-on video all about it so that we have the time to go through some examples</div>

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<div class="cue style-scope ytd-transcript-body-renderer">In the meantime, for more information on the story of quaternions and their relation to orientation in 3d space,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Quanta, a mathematical publication I'm sure a lot of you are familiar with,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">just put out a post in a kind of loose conjunction with this video. (link in the description)</div>

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<div class="cue style-scope ytd-transcript-body-renderer">If you enjoyed this, consider sharing it with some friends,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and if you felt like the narrative structure here was actually helpful for understanding</div>

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<div class="cue style-scope ytd-transcript-body-renderer">maybe reassure those friends who would be turned off by a large timestamp that good math is actually worth the time.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And many thanks to the patrons among you. I actually spent way longer than I care to admit on this project,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">so your patience and support is especially appreciated this time around.</div>

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<div id="label-text" class="style-scope yt-dropdown-menu">English</div>

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Visualizing quaternions (4d numbers) with stereographic projection

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