![]() ![]() Of course we can continue this line of thought: 4-dimensional space, for a mathematician, is identified with the sets of quadruples of real numbers, such as (5,6,3,2). This procedure extends to all higher dimensions. Of course this does not answer the physicist's question, of whether such dimensions have any objective physical existence. But mathematically, at least, as long as you believe in numbers, you don't have much choice but to believe in 4-dimensional space too. Well that is fine, but how can such spaces be imagined? What does the lair of Yog-Sothoth actually look like? This is a much harder question to answer, since our brains are not wired to see in more dimensions than three. But again, mathematical techniques can help, firstly by allowing us to generalise the phenomena that we do see in more familiar spaces.Īn important example is the sphere. If you choose a spot on the ground, and then mark all the points which are exactly 1cm away from it, the shape that emerges is a circle, with radius 1cm. If you do the same thing, but in 3-dimensional space, we get an ordinary sphere or globe. Now comes the exciting part, because exactly the same trick works in four dimensions, and produces the first hypersphere. What does this look like? Well, when we look at the circle from close up, each section looks like an ordinary 1-dimensional line (so the circle is also known as the 1-sphere). The difference between the circle and the line is that when viewed from afar, the whole thing curves back to connect to itself, and has only finite length. In the same way, each patch of the usual sphere (that is to say, the 2-sphere) looks like a patch of the 2-dimensional plane. Again, these patches are sewn together in a way that leaves no edges, and has only finite area. So far, so predictable, but exactly the same thing is true for the first hypersphere (or 3-sphere): each region looks just like familiar 3-dimensional space. We might be living in one now, for all we can see. But just like its lower dimensional cousins, the whole thing curves around on itself, in a way that flat 3-dimensional space does not, producing a shape with no sides, and only finite volume. From geometry to topology to differential topology Of course we do not stop here: the next hypersphere (the 4-sphere), is such that every region looks like 4-dimensional space, and so on in every dimension. “Surface Area of a Sphere.” Math Fun Facts.Like geometry, topology is a branch of mathematics which studies shapes. See also Volume of a Ball in N Dimensions. So, if I tell you the 4-dimensional “volume” of the 4-dimensional ball is (1/2)*Pi 2*R 4, what is 3-dimensional volume of its boundary? But the derivative is approximately the change in ball volume divided by (delta R), which is thus just (surface area of the sphere). The spherical shell's volume is thus approximately (surface area of the sphere)*(delta R). For the ball, a small change in radius produces a change in volume of the ball which is equal to the volume of a spherical shell of radius R and thickness (delta R). Let your students tell you those geometry formulas if they remember them. Similarly, the volume of a ball enclosed by a sphere of radius R is (4/3)*Pi*R 3.Īnd the formula for the surface area of a sphere of radius R is 4*Pi*R 2.Īnd, you can check that the latter is the derivative of the former with respect to R. The formula for the circumference of a circle of radius R is 2*Pi*R.Ī simple calculus check reveals that the latter is the derivative of the former with respect to R. The area of a disk enclosed by a circle of radius R is pi*R 2. ![]()
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