Every Complex Geometry Shape Explained

serinsky triangle take three identical equilateral triangles and join them at the vertices so that they form another equilateral triangle in the middle then shrink this shape down by factor of 1/2 take three identical copies of it and join them in similar way if you do this process over and over again the shape you end up approaching is called the serinsky triangle named after polish mathematici vatwa serinsky the serinsky triangle is an example of fractal which is essentially shape that has infinite Detail no matter how far you zoom in it never Smooths out in particular this Pinsky triangle is self similar fractal it is composed of three smaller copies of itself the study of fractals also gives rise to the idea of fractal Dimension line segment is considered one-dimensional square is considered two-dimensional and cube is considered three-dimensional if you scale up the dimensions of each of these by factor of two then the line segment's length is scaled by 2 to the first power or two the Square's area is scaled by 2^ 2ar or 4 and the Cube's volume is scaled by 2 cubed or 8 in each case the exponent is equal to the dimensionality of the object if you scale up the dimensions of the cinsky triangle by factor of two it becomes three times as large as result its dimensionality can be found by solving for in the equation 2 ^ of = 3 this equation can be solved with logarithms taking the logarithm base 2 of both sides we see that equal log base 2 of 3 which is about 1.58 in this sense the cinsky triangle is approximately 1.58 5 dimensional this is called its house dorf Dimension named after German mathematician Felix house dorf Tesseract the Tesseract is the four-dimensional analog of the cube just as line segment is formed by connecting two points square by connecting four line segments and cube by connecting six squares TCT is formed by connecting eight cubes these eight cubes are called the facets of the test ract for each Dimension the analog of the cube is known as the n-dimensional hyper Cube obviously four-dimensional shapes are difficult to visualize in world with only three dimensions of space however one option is to look at its 3D projection just says the 2D projection of 3D object can be thought of as its 2D shadow the 3D projection of 4D object can be thought of as its 3D Shadow just like the cube has volume and surface area Tesseract has 4D hyper volume and surface volume for side length the length of line segment is the area of square is squ and the volume of cube is cubed accordingly the hyper volume of tesseract is is to the 4th power the surface area of cube is obtained by adding together the areas of its six square facets yielding 6 s² likewise the surface volume of tesseract is obtained by adding together the volumes of its 8 cubic facets yielding 8s Cub Klein bottle let's start with mobia strip named after German mathematician August Ferdinand Mobius this is standard object that you could make at home just take paper strip give one end half twist and attach the ends together this results in piece of paper with just one side the mobia strip is something called nonorientable surface meaning that clockwise and counterclockwise rotation cannot be distinguished within it if you imagine yourself traveling along the length of the mobia strip Upon returning to your starting point you would find yourself upside down from your starting orientation of course this supposes that you're 3D object traveling on top of the mobia strip but if you're actually 2d object living within the mobia strip then traveling along it back to your starting position would cause you to become your mirror image accordingly rotations that once looked clockwise would now look counterclockwise so an orientation cannot be consistently defined for this surface the Klein bottle named after German mathematician Felix Christian Klein is another example of non-orientable surface however it has no boundary meaning there are are no points where the surface abruptly stops it does not intersect itself though it often seems to in visualizations due to the limitations of 3D space in 4D space it is easily constructed from the mobia strip just take two copies of the mobia strip and glue their edges together or in the words of Austrian Canadian mathematician Leo Moser mathematician named Klein thought the Mobius band was divine said he if you glue the edges of two you'll get weird bottle like mine mandelbrot set the mandelbrot set named after French American mathematician Beno mandelbrot arises in the study of complex numbers We Begin by picking some number in the complex plane using we will Define complex function called Sub with the equation Sub of equal z^2 + basically this function takes in number multiplies it by itself adds and spits out the result start by evaluating this function at zal 0 for instance if you chose = 1 then we have sub 1 of 0 = 0^ 2 + 1 = 1 then take the result and plug it back into the same function here that would give you sub 1 of 1 = 1^ 2 + 1 = 2 keep doing this over and over again depending on the value you chose for the resulting sequence of numbers may stay bounded in absolute value or it may diverge toward Infinity in our equals one case it diverges since the sequence goes 0 1 2 5 26 and so on the Mandel BR set is the set of all possible values of you could choose that result in Ed sequence as we saw the number one is therefore not an element of the mandal BR set but the number ne1 is since that produces the bounded sequence 01 Etc the mandal BR set is contained entirely within the disc of radius 2 centered at the origin and its boundary has house dwarf Dimension 2 if you draw the mandal BR set in the complex plane you get very intricate shape infinitely intricate in fact making it fractal zooming in we see that it is self similar fractal at certain points and variety of other patterns can be found due to its intricacy the mandelbrot set has been cited as an example of mathematical Beauty particularly how complex patterns can arise from simple definitions vastra function in calculus you may know about the concept of differentiable function if you take the graph of function and zoom zo in at certain point it may straighten out looking more and more like line if it does then the line being approximated is called the tangent line if this line is not vertical then the function is differentiable at that point and the derivative is the slope of the tangent line in order for function to be differentiable at point it must be continuous there meaning you can draw its graph without picking up your pencil also it cannot bend sharply there for example the absolute value function is not differentiable at zero no matter how far you zoom in on its graph at equal 0 it never straightens out it is easy to imagine continuous function that is non-differentiable at finite or even countably infinite amount of points since that just means that graph has sharp Bend there however it is much harder to imagine continuous function that's infinitely Jagged and doesn't smoothen out anywhere thus for long time mathematicians assumed that there was no function that is continuous everywhere and differentiable nowhere the Strauss function is function that is continuous everywhere and differentiable nowhere it was discovered by German mathematician Carl VRA and first published in 1872 vuss defined it using an infinite sum called Fier series here must be strictly between 0 and 1 must be positive odd integer and * must be greater than 1 plus + 32 Pi whatever values you pick for and that follow these rules you will get the exact same resulting function the graph of the Strauss function is self similar fractal curve however such curves were hard to visualize back then the existence of the Strauss function destroyed several proofs that relied on continuous functions being differentiable almost everywhere so it was denounced by mathematicians later on mathematicians apparently came to the realization that counterintuitive facts can be true and the virus st's function is widely accepted today Zyer surface take long piece of rope and arrange it into whatever form you like then glue together the two dangling ends mathematically the resulting object is known as knot if you can stretch and squish one knot into another knot without using self intersections then the two knots are the same if you take combination of one or more knots where these knots may or may not be separable you get an object called link knots and links are important objects of study in the mathematical field of knot Theory the simplest knot is the unot which is basically just loop of rope without anything being tied the next simplest knot is the troil knot which can be created by connecting the ends of an overhand knot as for links an unlink is any finite collection of circles that aren't connected together at all the Huff link named after German mathem ition fron hop consists of two circles linked together and the baroman Rings named after the aristocratic bomo family are three linked circles that fall apart if one is destroyed Zyer surface named after German mathematician Herbert Zer is an orientable surface with boundary of knot or link the simplest example is disc which is Surface whose boundary is circle which is just an unot noting the requirement of orientability in the definition the mobia strip is not Zer surface even though its boundary is an unot zipper surface formed by hop link may be resemblant of mobia strip but it is actually topologically equivalent to an annulus the plain region bounded by two concentric circles and is thus orientable finally zher surface formed by the baroman Rings looks like this