![]() We will publish a gallery of the curves our panel considers the most beautiful. Submit the equations for your two or three most beautiful curves as comments to this column. Try to come up with the most pleasing curves you can. Go wild! It’s like having a magic Spirograph whose pen never slips. ![]() You can use the insights you gained from considering question 1 above - or not. Now start playing with the parameters - change the coefficients, fiddle with the signs, do anything you want using sine and cosine terms. The mystery is this: What do the coefficients 6 and 14 have to do with generating fivefold symmetry? Why did we use –14 instead of 14 in question 1? Can you explain how this formula works to a general audience? Question 3:Īnyone can generate the above curve - use your favorite graphing app, or enter the above expression into Wolfram Alpha or the excellent graphic calculator Desmos. Note that this is a parametric equation - the first half gives the x-coordinate and the second part the y-coordinate as the parameter t goes from 0 to 2π. ![]() Look at the family of symmetrical curves below, one of which is featured in Farris’ book as a “mystery curve”: ( Update: The solution is now available here.) Question 1: The puzzles below merely skim the surface of the connection between math and art that the book explores in depth, but they are designed to encourage all of our readers to create stunning visual patterns using mathematics. They are inspired by the more elementary parts of Farris’ book, which discusses different types of symmetries in wallpaper, friezes and Escher-esque morphing patterns using a host of mathematical techniques involving groups, vector spaces, Fourier series, rosette functions, wave functions and several others. I believe our ability to find beauty in both mathematics and art reveals something deep about the human mind, a topic we may explore in the solution column. His book uses the abstract, esoteric beauty of mathematical equations to create the universally appealing beauty of visual art. Vincent Millay, and Frank Farris shows us what that means. “Euclid alone has looked on Beauty bare,” wrote the poet Edna St. Since ball-and-stick models are often a favourite starting point for physicists and chemists interested in 3D graphics, I've collected the formulas needed to do the above with cylinders (without end caps, with flat end caps, or with spherical end caps) to my Wikipedia user page.Lavished with many beautiful illustrations of this kind, Farris’ book is a joyful yet serious exploration of how mathematics can create beautiful patterns and provide us with a deep understanding of symmetry, one of the underlying principles of great art. In general, pick the sign that yields the smaller, but positive, $R$. If we are outside the sphere, use $-$ above if we are inside the sphere, use $+$ above. Therefore, the 2D coordinates of that detail on the window are Let's say one of the 3D coordinates of an interesting detail, say a corner of the greenish cube above, are $(x, y, z)$. In a very real sense, those coordinates are obtained by linear interpolation, except that one end of the line segment is at the eye (which we already decided is the origin, so coordinates $(0, 0, 0)$, the other end is at the 3D coordinates of the detail we wish to project, and the interpolation point is where that sight line (usually called "ray") intersects the view plane (the window, in our case). The blue pane is the window, the eye is at the lower left corner, and we are interested in the projected coordinates (projected to the window, that is) of the four corners of some cube at some distance. Here is a rough diagram of the situation: These coordinates are what OP needs to draw 3D pictures to a 2D surface. If we know the 3D coordinates in the above coordinate system of interesting details outside, 3D projection tells us their coordinates on the surface of the window. Thus, the center of the window is at $(0, 0, d)$, where $d$ is the distance from the eye to the window. Using OP's conventions, $x$ axis increases up, $y$ axis right, and $z$ axis outside the window. If you stand in the center of the window, looking out through the center of the window, then we can treat the center of your eye (more precisely, the center of the lens in the pupil of your dominant eye) the origin in 3D coordinates. Let's assume you stand in front of a window, looking out. It is all based on optics, and (linear) algebra. The hard part is understanding how it is done and that is what I shall try to explain here.
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