Visualizing 4D Geometry - A Journey Into the 4th Dimension [Part 2]

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this is not flatland this is actually our world and right here is our home earth but why does it look so flat before the flat earthers out there start getting excited I'm certainly not saying the earth is flat what I've done here is I've taken our 3d coordinate system and I've squashed it to make room for a fourth dimension aw axis so every horizontal flat plane that we see in this representation of boarded space is actually a three-dimensional hyper plane and our Flat Earth is actually not so flat if we look at it in 3d space but where exactly is this well this is our reality and as three-dimensional creatures our W coordinate is zero just like the Z coordinate of a flatlander is zero so everything that we see interact with in our world exists on this W equals zero hyperplane and so we will never be able to physically see what a four-dimensional object looks like because all we will be able to see is a 3-dimensional cross-section or a slice of a 4d object as it passes through this hyper plane this is actually the same problem that the Flatlanders had they can never physically see what a three-dimensional object looks like only a slice as it passes through their two-dimensional plane but don't get discouraged because by studying these slices it's possible to build an understanding of what 4d geometry looks like in a way that can supersede the need for direct visualization so first let's begin by looking at 4d objects from a more mathematical point of view consider the formulation of a circle we can construct a circle algebraically using the Pythagorean theorem and of course we can change its location and size by changing its centre coordinates and radius values but that's only in two dimensions in three dimensions we have an additional z axis to consider and we account for this by using the Pythagorean theorem in three dimensions which results in an additional Z squared term in our equation and four dimensions it's really not much different by adding a w squared term to our equation we could transform our sphere into a four-dimensional hyper sphere and we can use this equation to determine what a three-d slice would look like if this hyper sphere were to pass through our world since everything that we see is on the W equals zero hyperplane let's substitute W equals zero into our equation simplifying gives us this form which we recognize as the equation for a sphere so math tells us that 3d slices of a hyper sphere are spheres and the size of these spherical slices depend on the hyper planes W not coordinate and for T space so back to the representation of for T space we can draw our four dimensional hyper sphere like this and as it descends down the W axis eventually it intersects the W equals zero hyper plane where it enters our reality allowing us to see a three-dimensional slice by watching the hypersphere passed through our world we can visualize this geometry by watching how these 3d slices change when a three-dimensional sphere intersects flatland it creates a growing and shrinking circle caused by the roundness of the sphere in the third dimension Flatlanders can compile these slices in their mind and visualize its three-dimensional roundness by observing the rate of change of the radius similarly as the hyper sphere moves into and out of our world it shows up as spheres that grow and shrink according to the curvature in the fourth dimension so right now by watching the morphing that's going on we are actually in our limited perception seeing four-dimensional roundness okay so what would a hyper color look like if it were to pass through our world again we can start by analyzing its algebraic formulation by definition the boundary of a 3d cone is formed from an extrusion of a circular base to a point in the Z direction and and for D the hyper cone is an extrusion from a spherical base to a point in the W direction which is also why it's sometimes referred to as the spherical cone and of course we will only see a slice given by the intersection with the W equals zero hyperplane and so our equation tells us that as long as the axis of the hyper cone is aligned with the W axis each slice is a sphere whose radius depends on the W coordinate of the hyper come so intersecting our world would look something like this [Music] but what would we see if the hyper cone rotated through four dimensional space like this well first let's consider just a single arbitrary orientation obviously this won't be a sphere since the slice is not aligned with the axis of the cone but we can still figure out what this 3d slice looks like by reconstructing it piece by piece let's go ahead and break up the three dimensional intersection region into two dimensional pieces now remember our equation tells us that slice is taken along the axis of the hyper cone our spheres and so each piece is actually a two-dimensional portion of the three dimensional spherical slice of the four dimensional hyper cone so the object that we would see if the hyper cone intersected our world at this angle would be the region that is being swept out by the red slices [Music] you which in this case is something called a paraboloid a 3d version of the parabola [Music] this hyper cone can also make other fun shapes for example consider this orientation this 3d slice of the hyper cone is an ellipsoid a 3d version of the ellipse or how about the hyperboloid a 3d version of the hyperbola at this point we should start seeing an interesting pattern care if we take two-dimensional slices of a normal 3-dimensional cone we get things called conics these include circles ellipses parabolas and hyperbolas depending on the angle that the cone is sliced but if we take three dimensional slices of a four-dimensional hyper con we get the 3d analogs to these 2d comics the spheres ellipsoids paraboloids and hyperboloids and the reason this happens is because of the hyper cone spherical symmetry around the W axis so to get back to the original question a hyper cone moving and rotating through four dimensional space we look like a smooth morphing between the different 3d conic sections and as long as it intersects at reality we would see some pretty cool shapes that would certainly be unexplainable with only a three dimensional worldview [Music] [Music] now I want to show you a four-dimensional hypercube if you're a mathematician you may know this by another name the for cube or if you've seen the movie interstellar you probably know it as the tesseract so to construct a tesseract we can start with a point and extrude it in the x-direction to create a line then we can take that line and extrude it in the y direction to create square and then we can extrude that square in the z direction to make a cube and finally we extrude that cube perpendicular to itself in the W direction to make the tesseract so we're a square has one dimensional line boundaries and the cube has two dimensional square boundaries a tesseract has three dimensional cube boundaries called cells in fact there are exactly eight cells that make up its boundary so in order to determine what a 3-dimensional slice of the hypercube looks like we need to find the intersection of our hyperplane with each of the eight cube cells that make up the tesseract boundary since an intersection of a plane with a cube can only be a triangle quadrilateral Pentagon or a hexagon we only need to find the points of intersection of the W equals zero hyperplane with the edges of a single cube cell then we connect the dots to make a polygon and repeat this for all eight cells and the polygons will come together to form a closed boundary that defines the three-dimensional slice so this is a three-dimensional slice of our tesseract that is halfway intersecting our reality and we can rotate it in any combination of the X Y X Z and Y Z planes as we can any other three-dimensional object but nothing interesting happens in fact you're probably thinking that I'm lying to you that this actually isn't a four-dimensional object at all but the reason it looks so normal is because we are only rotating it in X Y Z space meaning that the W coordinates of every point and the tesseract are not changing at all it's kind of like a cube rotating at flatland when rotations are restricted to only the XY flatland plane the Z coordinates don't change so it will always look like a square and the Flatliners would agree that it looks pretty normal however in four dimensions there are not three planes of rotation but six planes and the visuals get pretty interesting when we start rotating in the XW YW + ZW planes so this is not exactly what we'd expect to see from a rotation it's almost as if the hypercube is just stretching and contracting and magically changing colors but let's see what's really going on behind the scenes in a 2d world if we looked at a cube rotating an XE plane we would see the same stretching and contracting and color changes but what's really happening isn't very magical it's actually quite clear the stretching and contracting is due to the fact that the diagonal is longer than the side length and the color changes are just a result of edges of the cube rotating through the plane [Music] for the tesseract it's really the same thing but one dimension higher stretching and contracting is still from the diagonal being longer than the side length but the sudden color changes in the faces are a result of entire phases of the tesseract rotating through our hyperplane things get even stranger when we start rotating around to planes [Music] [Music] again we can make sense of what's happening by looking at the 2d case the squared distorts and magically grows two additional sides but this is really due to the corners of the cube rotating through the plane [Music] and 3d the additional faces that we see our result of edges of our test rack rotating through our hyperplane there's also a special subset of two plane rotations called double rotations or Clifford displacements and these are special because they represent rotations around two orthogonal planes which is pretty cool considering this is impossible to do in only three dimensions we can also rotate our tesseract in any three planes at once and again we can see a 2d analog of the three plane rotation to gain some insight into what we are seeing [Music] you [Music] we can also rotate around more than three planets additionally we can learn more about the geometry of the tesseract by orienting it in certain ways and studying 3d slices as we move it along the W axis into and out of our world we will be taking a look at the cases when the tesseract is oriented so first face first edge first and corner first and to help understand and visualize what we see before each of these cases I am going to show you the 2-dimensional flatland analog and what we'll notice is that since the tesseract has four perpendicular edges coming out of each corner and the cube only has three the 3d slices of the tesseract will look just like the 2d slices of the cube except just one dimension higher so here we go [Music] [Music] [Music] [Music] [Music] [Music] [Music] you [Music] [Music] [Music] four dimensional geometry is actually quite beautiful partly due to its apparent complexity but also because it's filled with mystery and other worldliness because everything and everyone that we know lived right here at W equals zero our four dimensional address and as far as we know no one can ever leave we are confined trapped even to this hyper plane we are the Flatlanders of a four dimensional world and infinitely thin insignificant slice of something else is higher dimensional reality whether you find that comforting or chilling is up to you but it opens the door to a very real question what if our experience of the physical world is merely a representation of something extra dimension and that our understanding of reality is limited to a three dimensional interpretation that is inconsistent with actuality maybe there's a whole new world out there that we have yet to discover a world that extends beyond into the fourth dimension new space new things maybe even new life but all of us here watching this video are the pioneers of this new place the modern-day explorers from the history books continuing the age long search for answers to the mysteries of the world that we live in and the universe that's surrounded [Music] until next time [Music] hey guys have you made it all the way to the end of this video be sure to leave a comment and let me know what kind of four-dimensional stuff that you'd like to see it covered in future videos of the series after all this is a series dedicated to exploring the fourth dimension and I know a lot of you guys have some really cool insights and ideas to share so I'm excited to read those in the comes down below and anyway thanks for watching and I will see you guys next time

Info

Channel: The Lazy Engineer

Views: 283,144

Rating: 4.9534645 out of 5

Keywords: journey into the 4th dimension, fourth dimension, 4th dimension, what is the 4th dimension, hypersphere, hypercone, hypercube, tesseract, 3d slice of 4d, visualizing 4d, documentary, 4d toys, 4d game, 4th dimension explained, fourth dimension explained, the lazy engineer, 4d hypersphere, 4d hypercone, what does, look like, how to see in 4d, 4d hypercube, interstellar tesseract

Id: 4URVJ3D8e8k

Channel Id: undefined

Length: 20min 0sec (1200 seconds)

Published: Sat Aug 12 2017

Reddit Comments

Thanks OP, this was not only an incredibly clear explanation of the concept of 4 dimensional objects, but also relaxing as hell, needed something like this to end my day

👍︎︎ 6 👤︎︎ u/Ololondo 📅︎︎ Jan 07 2019 🗫︎ replies

I can't even visualize 2D geometry

👍︎︎ 3 👤︎︎ u/Powerspawn 📅︎︎ Jan 07 2019 🗫︎ replies

Is it possible that the expansion of the universe is actually a 4d universe passing through our 3D perception. If so, can we figure out the shape of that universe?

👍︎︎ 2 👤︎︎ u/RedEbw0307 📅︎︎ Jan 07 2019 🗫︎ replies

I’m just reading Death’s End by Cixin Liu (3rd of the Remembrance of Earth’s Past trilogy) and there’s a large section of Part 2 where Cixin Liu discusses the crew of a spaceship finding and interacting with the 4th Dimension. Some similarities in this video. A very cool read!

👍︎︎ 1 👤︎︎ u/Markuu6 📅︎︎ Jan 07 2019 🗫︎ replies

The thumbnail looks like a surreal meme

👍︎︎ 1 👤︎︎ u/CriticalThaumaturgy 📅︎︎ Jan 07 2019 🗫︎ replies