WEBVTT 00:00:00.000 --> 00:00:03.379 Welcome, everyone, to today's deep dive. I'm 00:00:03.379 --> 00:00:05.099 really excited about this one. Yeah, it's going 00:00:05.099 --> 00:00:07.200 to be a fun one. It really is. Because our mission 00:00:07.200 --> 00:00:11.039 today is to understand this mathematical magic 00:00:11.039 --> 00:00:13.339 trick. It's a trick that lets computers solve 00:00:13.339 --> 00:00:15.900 wildly complex problems without actually having 00:00:15.900 --> 00:00:19.059 to do impossibly complex math. Which I know sounds 00:00:19.059 --> 00:00:21.420 like a total contradiction. Right. And we are 00:00:21.420 --> 00:00:24.199 pulling our insights today primarily from the 00:00:24.199 --> 00:00:27.140 Wikipedia article on the kernel method in machine 00:00:27.140 --> 00:00:30.859 learning. I know the term kernel method might 00:00:30.859 --> 00:00:32.799 sound intimidating. It sounds like something 00:00:32.799 --> 00:00:34.920 buried in an advanced computer science textbook. 00:00:35.320 --> 00:00:37.759 It definitely does. But for you listening, this 00:00:37.759 --> 00:00:40.579 deep dive is really the ultimate shortcut to 00:00:40.579 --> 00:00:43.539 understanding how algorithms categorize the messy 00:00:43.539 --> 00:00:46.320 real world we live in. It's perfect for anyone 00:00:46.320 --> 00:00:49.000 who wants to grasp high level AI concepts, but 00:00:49.000 --> 00:00:52.000 doesn't want to get a PhD in calculus just to 00:00:52.000 --> 00:00:54.219 understand it. Exactly. You don't need the PhD 00:00:54.219 --> 00:00:56.579 today. Good. Because I don't have one. Yeah. 00:00:56.719 --> 00:00:58.840 And to help us navigate this, I have our expert 00:00:58.840 --> 00:01:01.380 guide here to translate all that dense mathematics 00:01:01.380 --> 00:01:04.859 into some clear insights. I am ready. And I think 00:01:04.859 --> 00:01:07.120 to really appreciate the brilliance of this magic 00:01:07.120 --> 00:01:09.739 trick, we first need to define the fundamental 00:01:09.739 --> 00:01:12.180 problem that machine learning algorithms face 00:01:12.180 --> 00:01:15.159 when they categorize data. Right. So where do 00:01:15.159 --> 00:01:17.200 we start? We start with something called pattern 00:01:17.200 --> 00:01:20.299 analysis. Pattern analysis. OK. Yeah. It's the 00:01:20.299 --> 00:01:23.879 general task of finding clusters. rankings, principal 00:01:23.879 --> 00:01:26.959 components, and classifications in big data sets, 00:01:27.519 --> 00:01:30.459 like trying to find order in the chaos. So like 00:01:30.459 --> 00:01:33.019 figuring out if an email is spam or not. Exactly. 00:01:33.120 --> 00:01:36.500 Or if an image is a cat or a dog. And historically, 00:01:36.920 --> 00:01:39.959 to do this, engineers use what are called linear 00:01:39.959 --> 00:01:42.739 classifiers. OK, linear classifiers, meaning 00:01:42.739 --> 00:01:46.340 lines. Literally lines, yes. A linear classifier 00:01:46.340 --> 00:01:48.519 attempts to draw a straight boundary to separate 00:01:48.519 --> 00:01:51.150 different classes of data. OK. They're heavily 00:01:51.150 --> 00:01:53.370 used because they're mathematically simple, just 00:01:53.370 --> 00:01:56.450 basic addition and multiplication. But, and this 00:01:56.450 --> 00:01:59.170 is the big, but they struggle heavily with nonlinear 00:01:59.170 --> 00:02:01.129 problems. Okay, let's unpack this because I want 00:02:01.129 --> 00:02:03.489 to make sure you, the listener, can really visualize 00:02:03.489 --> 00:02:06.370 this. Imagine you have a bowl mixed with red 00:02:06.370 --> 00:02:08.830 and blue marbles. Okay, I'm picturing it. And 00:02:08.830 --> 00:02:11.430 a linear classifier is like trying to separate 00:02:11.430 --> 00:02:13.969 the red ones from the blue ones by sliding a 00:02:13.969 --> 00:02:16.629 single flat stiff piece of cardboard into the 00:02:16.629 --> 00:02:18.870 bowl. Right. Which works great if all the red 00:02:18.870 --> 00:02:20.330 ones are on the left and the blue ones are on 00:02:20.330 --> 00:02:22.889 the right. Exactly. The cardboard slides right 00:02:22.889 --> 00:02:26.189 down the middle. But what if the data is messy? 00:02:26.789 --> 00:02:29.509 What if the red marbles are in a clump in the 00:02:29.509 --> 00:02:32.430 very center, completely surrounded by a ring 00:02:32.430 --> 00:02:34.930 of blue ones? Then you have a serious problem. 00:02:35.069 --> 00:02:37.370 Right, because a single flat plane can't separate 00:02:37.370 --> 00:02:39.289 them. You'd slice right through the blue ring 00:02:39.289 --> 00:02:42.770 to get to the red center. So my pushback here 00:02:42.770 --> 00:02:46.719 is if the data is messy, Why not just build a 00:02:46.719 --> 00:02:49.360 bendy complex piece of cardboard? It's a totally 00:02:49.360 --> 00:02:51.780 logical question. Why not just draw a circle 00:02:51.780 --> 00:02:54.400 around the red marbles? Yeah. Why does it have 00:02:54.400 --> 00:02:57.000 to be a straight piece of cardboard? Well, the 00:02:57.000 --> 00:03:00.439 problem is that in raw representation, data usually 00:03:00.439 --> 00:03:02.900 has to be explicitly transformed to make sense 00:03:02.900 --> 00:03:05.409 of it. Engineers use what's called a feature 00:03:05.409 --> 00:03:08.490 map to push the data into a feature vector representation. 00:03:08.949 --> 00:03:10.770 Okay, that's a lot of jargon. What does that 00:03:10.770 --> 00:03:13.810 mean for our marbles? It means we are assigning 00:03:13.810 --> 00:03:16.889 brand new mathematical coordinates to every single 00:03:16.889 --> 00:03:19.750 marble to move them into a new space where a 00:03:19.750 --> 00:03:21.969 flat piece of cardboard will actually work. Oh 00:03:21.969 --> 00:03:24.409 wow, so we warp the space instead of bending 00:03:24.409 --> 00:03:27.150 the cardboard. Exactly, like adding a third dimension, 00:03:27.789 --> 00:03:31.039 a z -axis for height. based on how far a marble 00:03:31.039 --> 00:03:33.300 is from the center. Oh, I see. So the red marbles 00:03:33.300 --> 00:03:35.900 in the center stay low, but the blue ones on 00:03:35.900 --> 00:03:38.699 the outside get mapped up high. Right. The flat 00:03:38.699 --> 00:03:41.280 bowl becomes a 3D funnel. And then you can just 00:03:41.280 --> 00:03:43.719 slide your flat cardboard horizontally between 00:03:43.719 --> 00:03:46.000 the low red marbles and the high blue ones. That 00:03:46.000 --> 00:03:49.020 is brilliant. It is. But what's fascinating here 00:03:49.020 --> 00:03:53.000 is that explicitly doing this math is incredibly 00:03:53.000 --> 00:03:55.639 taxing. Taxing how? I mean, if you just have 00:03:55.639 --> 00:03:58.379 50 marbles and one new dimension, Your laptop 00:03:58.379 --> 00:04:00.740 does it instantly. But what if you have millions 00:04:00.740 --> 00:04:03.060 of high -resolution images and you need to add 00:04:03.060 --> 00:04:05.479 thousands of new dimensions? Oh, right. Calculating 00:04:05.479 --> 00:04:07.780 all those new coordinates for every single data 00:04:07.780 --> 00:04:10.159 point. Across thousands of dimensions, exactly. 00:04:10.439 --> 00:04:13.360 It requires astronomical computing power. The 00:04:13.360 --> 00:04:15.719 matrix of numbers gets so huge that the computer 00:04:15.719 --> 00:04:18.939 just runs out of memory and crashes. So explicitly 00:04:18.939 --> 00:04:21.199 building the Bendy Card board is just too much 00:04:21.199 --> 00:04:23.360 work. Way too much work. Which brings us to the 00:04:23.360 --> 00:04:25.970 workaround. Since explicitly transforming the 00:04:25.970 --> 00:04:29.410 data is too hard, the algorithms use this elegant 00:04:29.410 --> 00:04:32.629 mathematical workaround. Yes, the kernel trick. 00:04:32.850 --> 00:04:35.329 The kernel trick! I just, I love that name. It's 00:04:35.329 --> 00:04:37.689 great, right. The kernel trick is the hero here. 00:04:38.230 --> 00:04:41.170 It solves the nonlinear problem without all that 00:04:41.170 --> 00:04:43.410 heavy lifting. Okay, so how does the trick actually 00:04:43.410 --> 00:04:46.550 work? Well, a kernel is simply a similarity function 00:04:46.550 --> 00:04:49.920 over pairs of data points. And it computes this 00:04:49.920 --> 00:04:52.399 similarity using something called inner products. 00:04:52.699 --> 00:04:54.560 Inner products. Yeah, conceptually, an inner 00:04:54.560 --> 00:04:56.699 product is just a mathematical way to get a single 00:04:56.699 --> 00:04:59.899 number that represents how much two things overlap, 00:05:00.019 --> 00:05:02.019 how similar they are. OK, so just a similarity 00:05:02.019 --> 00:05:05.589 score. Exactly. And by using this trick. the 00:05:05.589 --> 00:05:08.670 algorithm can operate in a high dimensional implicit 00:05:08.670 --> 00:05:11.550 feature space. Wait, implicit? Implicit, that's 00:05:11.550 --> 00:05:14.110 the magic word. We operate in that complex space 00:05:14.110 --> 00:05:16.689 implicitly rather than explicitly. Here's where 00:05:16.689 --> 00:05:18.850 it gets really interesting because the source 00:05:18.850 --> 00:05:21.930 text mentions that this feature map can be infinite 00:05:21.930 --> 00:05:24.170 dimensional. It can, yeah. But wait, you're telling 00:05:24.170 --> 00:05:26.790 me that instead of drawing a complicated line 00:05:26.790 --> 00:05:30.589 in 3D, we are mapping data to infinite dimensions? 00:05:31.029 --> 00:05:34.750 How is that less work? That sounds like a computational 00:05:34.750 --> 00:05:36.990 nightmare. It completely sounds like a paradox. 00:05:37.290 --> 00:05:39.550 I mean, you just said adding a few thousand dimensions 00:05:39.550 --> 00:05:42.910 would crash a computer. I did. But here is the 00:05:42.910 --> 00:05:45.730 brilliance of the trick. We never actually compute 00:05:45.730 --> 00:05:48.769 the coordinates in that infinite space. We don't. 00:05:48.850 --> 00:05:51.959 No. We never go there. We only compute the inner 00:05:51.959 --> 00:05:55.379 products, the similarity between the images of 00:05:55.379 --> 00:05:58.240 all pairs of data. Oh, wow. Think of it like 00:05:58.240 --> 00:06:00.800 comparing the recipe cards for two incredibly 00:06:00.800 --> 00:06:03.980 complex cakes instead of actually baking the 00:06:03.980 --> 00:06:06.720 infinite layers of both cakes to taste the difference. 00:06:06.860 --> 00:06:08.339 OK, that makes so much sense. We just compare 00:06:08.339 --> 00:06:10.759 the raw ingredients. Exactly. The kernel function 00:06:10.759 --> 00:06:13.519 spits out a similarity score, and mathematically, 00:06:13.930 --> 00:06:17.290 that score perfectly corresponds to what the 00:06:17.290 --> 00:06:19.350 inner product would be if we had actually baked 00:06:19.350 --> 00:06:21.470 the infinite cake. So we get the result without 00:06:21.470 --> 00:06:23.389 doing the work. Right. And there is a strict 00:06:23.389 --> 00:06:25.430 mathematical foundation that allows this. It's 00:06:25.430 --> 00:06:27.589 called the Representer Theorem. Representer Theorem. 00:06:27.629 --> 00:06:30.110 Got it. It states that even though the feature 00:06:30.110 --> 00:06:33.569 map is infinite dimensional, finding the optimal 00:06:33.569 --> 00:06:36.550 flat plane to separate the data only requires 00:06:36.550 --> 00:06:39.509 a finite dimensional matrix from the user. So 00:06:39.509 --> 00:06:44.250 it's a massive computationally cheaper The ultimate 00:06:44.250 --> 00:06:46.350 shortcut. Okay, my mind is a little blown, but 00:06:46.350 --> 00:06:48.709 let's look under the hood. Because now that we 00:06:48.709 --> 00:06:51.310 know the trick involves bypassing the complex 00:06:51.310 --> 00:06:53.889 coordinates in favor of just measuring similarity, 00:06:54.449 --> 00:06:57.310 how does the machine actually use this to predict 00:06:57.310 --> 00:06:59.529 something it has never seen before? Well, kernel 00:06:59.529 --> 00:07:02.029 methods are what we call instance -based learners. 00:07:02.290 --> 00:07:04.829 Instance -based learners, as opposed to? As opposed 00:07:04.829 --> 00:07:06.910 to learning a generalized rule and throwing the 00:07:06.910 --> 00:07:09.410 data away. Kernel methods actually remember the 00:07:09.410 --> 00:07:11.490 training examples. Oh, they just memorize them. 00:07:11.629 --> 00:07:13.680 Basically, yes. They remember them and assign 00:07:13.680 --> 00:07:16.980 them weights. So when a new unlabeled input arrives, 00:07:17.300 --> 00:07:20.000 say a new image, the kernel compares it to each 00:07:20.000 --> 00:07:22.459 of the memorized training inputs. OK. And it 00:07:22.459 --> 00:07:24.680 computes a similarity score for every single 00:07:24.680 --> 00:07:28.240 pair. Then the machine computes a weighted sum 00:07:28.240 --> 00:07:30.600 of all those similarities to predict whether 00:07:30.600 --> 00:07:33.100 the new input gets a positive or negative label. 00:07:33.439 --> 00:07:36.420 So what does this all mean? Let me pitch an analogy 00:07:36.420 --> 00:07:39.240 to you, the listener. It's like you're trying 00:07:39.240 --> 00:07:41.259 to figure out if you'll like a new movie. Okay, 00:07:41.319 --> 00:07:43.560 I like where this is going. Instead of analyzing 00:07:43.560 --> 00:07:46.980 its script, its lighting, the director's history, 00:07:47.139 --> 00:07:49.620 which is like explicitly calculating its coordinates. 00:07:49.879 --> 00:07:53.069 Right. You just measure how similar this new 00:07:53.069 --> 00:07:55.829 movie is to your 10 favorite movies of all time, 00:07:56.329 --> 00:07:58.930 which are your training data. That is perfectly 00:07:58.930 --> 00:08:01.629 stated. You just sum up the similarity scores. 00:08:02.009 --> 00:08:04.370 If the total is positive, you'll like the movie. 00:08:04.490 --> 00:08:07.670 If it's negative, you skip it. Exactly. And the 00:08:07.670 --> 00:08:10.089 kernel function handles all that similarity measuring 00:08:10.089 --> 00:08:12.170 instantly. But wait, I have to push back here. 00:08:12.629 --> 00:08:15.509 Can we use any criteria for similarity? Can I 00:08:15.509 --> 00:08:17.250 just make up a random rule and say, you know, 00:08:17.370 --> 00:08:18.910 these are similar because they both have a red 00:08:18.910 --> 00:08:21.370 pixel? You could try, but the math would collapse. 00:08:21.529 --> 00:08:24.670 It would. Why? Because there are strict theoretical 00:08:24.670 --> 00:08:27.310 guardrails. Yeah. The main one is Mercer's theorem. 00:08:27.910 --> 00:08:30.370 Mercer's theorem. OK, what does that do? It dictates 00:08:30.370 --> 00:08:32.570 that the similarity function you choose must 00:08:32.570 --> 00:08:35.570 result in a Gram matrix or a kernel matrix that 00:08:35.570 --> 00:08:40.110 is positive semi -definite. Positive semi -definite. 00:08:40.110 --> 00:08:43.419 That sounds very dense. It is, yeah, but basically 00:08:43.419 --> 00:08:47.279 it's just the math's way of ensuring your similarity 00:08:47.279 --> 00:08:50.419 rules don't contradict the basic laws of geometry. 00:08:50.570 --> 00:08:54.009 Like saying city A is close to city B and B is 00:08:54.009 --> 00:08:56.970 close to C, but A and C are on opposite sides 00:08:56.970 --> 00:08:59.789 of the universe. Exactly. It prevents those impossible 00:08:59.789 --> 00:09:02.350 geometric contradictions so the algorithm can 00:09:02.350 --> 00:09:04.909 reliably find that flat plane. OK, that makes 00:09:04.909 --> 00:09:06.850 sense. But the text pointed out a fascinating 00:09:06.850 --> 00:09:09.529 real -world quirk about Mercer's theorem, didn't 00:09:09.529 --> 00:09:12.350 it? It did. And this is where theory meets real 00:09:12.350 --> 00:09:15.809 -world coding. Empirically, engineers found that 00:09:15.809 --> 00:09:18.009 even if a function doesn't perfectly satisfy 00:09:18.009 --> 00:09:20.509 Mercer's condition, it can still work reasonably 00:09:20.509 --> 00:09:23.529 well. Wait, really? Even if the geometry is technically 00:09:23.529 --> 00:09:26.110 broken? Yeah. As long as it merely approximates 00:09:26.110 --> 00:09:28.590 the intuitive idea of similarity, the algorithm 00:09:28.590 --> 00:09:31.090 can often still find a useful pattern. That is 00:09:31.090 --> 00:09:33.970 wild. The math is strict, but reality is a bit 00:09:33.970 --> 00:09:36.029 more forgiving. Exactly. It's a great example 00:09:36.029 --> 00:09:38.539 of a real -world heuristic. So having unpacked 00:09:38.539 --> 00:09:40.500 all this brilliant math and its guardrails, let's 00:09:40.500 --> 00:09:42.580 look at the real world impact, because this actually 00:09:42.580 --> 00:09:45.320 achieved a lot, right? Oh, it completely changed 00:09:45.320 --> 00:09:47.039 computer science. Tell me about the timeline 00:09:47.039 --> 00:09:49.779 here. So the underlying concept, the kernel perceptron, 00:09:49.840 --> 00:09:52.700 was actually described way back in the 1960s. 00:09:52.700 --> 00:09:57.000 OK. But the true golden age hit in the 1990s 00:09:57.000 --> 00:09:59.879 with the support vector machine. or SVM. Right, 00:10:00.120 --> 00:10:02.559 the SVM. When researchers injected the kernel 00:10:02.559 --> 00:10:05.080 trick into the SVM architecture, the results 00:10:05.080 --> 00:10:08.019 were staggering. They became highly competitive 00:10:08.019 --> 00:10:10.700 with neural networks. Which are like the heavyweight 00:10:10.700 --> 00:10:14.179 champions of AI. Exactly. Yeah. And they specifically 00:10:14.179 --> 00:10:16.559 crushed it in handwriting recognition. Oh, right. 00:10:17.019 --> 00:10:19.379 Recognizing handwritten digits is super messy. 00:10:19.559 --> 00:10:22.279 Everyone draws a seven differently. Very messy. 00:10:22.759 --> 00:10:26.320 But SVMs with kernel functions could implicitly 00:10:26.320 --> 00:10:29.159 map those messy pixels and draw crystal clear 00:10:29.159 --> 00:10:31.460 boundaries between a sloppy seven and a sloppy 00:10:31.460 --> 00:10:33.759 one. And from there, the applications just exploded. 00:10:34.120 --> 00:10:38.299 The article lists geostatistics, 3D reconstruction. 00:10:38.539 --> 00:10:40.779 The huge one. And bioinformatics, which I find 00:10:40.779 --> 00:10:43.539 incredible, like analyzing genetic data. Oh, 00:10:43.580 --> 00:10:45.600 bioinformatics is a stellar example. Because 00:10:45.600 --> 00:10:47.860 DNA is just massive strings of letters, right? 00:10:47.879 --> 00:10:49.840 But you can't draw a line through billions of 00:10:49.840 --> 00:10:53.480 A, C, G, and T's. You can't. Yeah. But by using 00:10:53.480 --> 00:10:56.519 specialized string kernels, scientists can implicitly 00:10:56.519 --> 00:10:59.659 map those DNA sequences and isolate gene clusters 00:10:59.659 --> 00:11:02.360 with incredible precision. It's phenomenal. But 00:11:02.360 --> 00:11:07.000 here, I spotted a catch in the source text. You 00:11:07.000 --> 00:11:10.120 found the fatal flaw. I did. The text says that 00:11:10.120 --> 00:11:12.139 this is slow to compute for data sets larger 00:11:12.139 --> 00:11:14.620 than a couple of thousand examples without parallel 00:11:14.620 --> 00:11:17.200 processing. It does say that. Wait. A couple 00:11:17.200 --> 00:11:21.480 of thousand. If this math is so elegant, why 00:11:21.480 --> 00:11:23.639 does it completely choke on just a few thousand 00:11:23.639 --> 00:11:27.419 data points? In the modern era of big data, a 00:11:27.419 --> 00:11:29.600 few thousand is basically nothing, right? If 00:11:29.600 --> 00:11:31.399 we connect this to the bigger picture, remember 00:11:31.399 --> 00:11:33.120 how we said it's an instance -based learner. 00:11:33.480 --> 00:11:36.289 Right. It remembers the training data. Exactly. 00:11:36.750 --> 00:11:38.990 So mathematically, the kernel function has to 00:11:38.990 --> 00:11:41.710 compute the similarity score between every new 00:11:41.710 --> 00:11:44.070 data point and every single piece of training 00:11:44.070 --> 00:11:47.049 data. Oh, no. And during training, it has to 00:11:47.049 --> 00:11:49.690 compare every training point against every other 00:11:49.690 --> 00:11:51.629 training point to build that gram matrix. Oh, 00:11:51.629 --> 00:11:54.289 wow. I see where this is going. Yet the complexity 00:11:54.289 --> 00:11:58.379 scales quadratically or even cubically. If you 00:11:58.379 --> 00:12:01.240 have just 100 ,000 training examples, which is 00:12:01.240 --> 00:12:05.240 tiny today, that matrix requires computing 10 00:12:05.240 --> 00:12:08.080 billion inner products. 10 billion calculations 00:12:08.080 --> 00:12:11.720 for a tiny data set. Exactly. The gram matrix 00:12:11.720 --> 00:12:14.320 just gets exponentially heavier. This is why 00:12:14.320 --> 00:12:16.320 parallel processing became a necessity. You have 00:12:16.320 --> 00:12:19.340 to split it across thousands of cores just to 00:12:19.340 --> 00:12:21.659 get an answer. Right. And it shows how tools 00:12:21.659 --> 00:12:24.159 have to fit their specific era and hardware. 00:12:24.440 --> 00:12:27.440 In the 90s, when datasets were smaller, kernel 00:12:27.440 --> 00:12:30.679 methods were king. Today, with petabytes of data, 00:12:31.019 --> 00:12:32.720 neural nets often take the lead because they 00:12:32.720 --> 00:12:34.539 scale better. Even if their internal math is 00:12:34.539 --> 00:12:37.299 less transparent. Exactly. Man, that is a fascinating 00:12:37.299 --> 00:12:40.799 evolution. So, to summarize the core aha moment 00:12:40.799 --> 00:12:43.779 for you listening. Kernel methods, and specifically 00:12:43.779 --> 00:12:46.299 the kernel trick, represent a real triumph of 00:12:46.299 --> 00:12:48.519 perspective. They really do. Instead of brute 00:12:48.519 --> 00:12:51.100 forcing a complex categorization problem in flat 00:12:51.100 --> 00:12:53.639 space, mathematicians found a way to jump into 00:12:53.639 --> 00:12:56.580 infinite dimensions using a shortcut based purely 00:12:56.580 --> 00:12:59.570 on similarity. And this raises an important question. 00:13:00.190 --> 00:13:02.710 Well, if the entire success of a kernel method 00:13:02.710 --> 00:13:06.049 relies on a human choosing the mathematical definition 00:13:06.049 --> 00:13:09.309 of similarity, the kernel function itself, what 00:13:09.309 --> 00:13:11.789 happens when human biases leak into that definition? 00:13:11.870 --> 00:13:14.330 Oh, wow. Right. We always assume math is entirely 00:13:14.330 --> 00:13:17.389 objective. But if we are the ones defining what 00:13:17.389 --> 00:13:20.070 makes two distinct points similar, are we just 00:13:20.070 --> 00:13:23.330 teaching the machine our own blind spots in infinite 00:13:23.330 --> 00:13:26.039 dimensions? That is a staggering thought. The 00:13:26.039 --> 00:13:28.139 math might be perfect, but the rules are still 00:13:28.139 --> 00:13:31.019 human. Exactly. Well, thank you so much for joining 00:13:31.019 --> 00:13:33.019 us on this deep dive. And for you listening, 00:13:33.299 --> 00:13:35.139 keep looking for those hidden shortcuts in the 00:13:35.139 --> 00:13:37.240 world around you. We will catch you on the next 00:13:37.240 --> 00:13:37.440 one.