The other well-known functions would be linear combinations of e^x, e^-x, sin x, and cos x.
The general solution looks like this: If you want the nth derivative to give you the original function, a solution is e^(Kx), where K is a constant such that K^n = 1. But there are n such (complex) values of K, and in general you take a constant multiple of each e^(Kx) and add them all together.
That includes the zero solution.
It may seem like sin x and cos x don’t fit here, but those are sums of multiples of e^(ix) and e^(-ix).
You also get sinh x and cosh x in much the same way.
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I tend to reserve “mathematician” for people who do research in the field as their occupation, and that isn’t me. I am merely a humble math nerd with a BS, who sometimes keeps up with research in type theory and some other areas that straddle the line between pure math and theoretical computer science.
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Earlier this month, Belgian mathematician and physicist Ingrid Daubechies received the US National Medal of Science for her work in wavelets, which apparently have important applications in data-compression.
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exp(x) = x^12 has countably infinitely many solutions in the complex numbers: twelve solutions for every branch of Lambert’s W function. Specifically, for every (k,n) where k is a twelfth root of unity and n is an integer, -12W_n(k/12) is a unique solution.
Is 12 the only number for which this is true?
No. The solutions would follow a similar pattern for any other positive integer exponent.
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Today is the 175th birthday of the first woman to earn a Ph.D. in Mathematics:
Also, 2025 is a square year and the only one this century. More 2025 facts can be found at:
Also, sparked by some comments in the Trivia thread, my mind has recently wandered back to something I’ve been calling hyper compass and straightedge construction, a generalization of compass and straight edge to 3-dimensions in which
A hyper compass, given points O and A, draws the sphere centered at O through point A.
A Hyper straight edge, givens non-collinear points A, B, and C, draws the plane through A, B, and C.
And of course, you still get lines between pairs of points, line segments, circular arcs, etc. but also the circles where a sphere and plane intersect, the polygons bound by coplanar points, spherical caps from planes cutting spheres, spherical polygons bound by points on a sphere, etc.
Presumably, all planar figures constructible with normal compass and straight edge are constructible with hyper compass and straight edge, but I find myself wondering if the third dimension allows for the construction of planar figures impossible with compass and straight edge alone(e.g. does three dimensions allow for angle trisection or cube rooting?), what of the platonic, kepler-poinsot, Archimedean, Catalan, uniform prisms and antiprisms, uniform star polyhedra, and Johnson solids can be constructed with such tools, which of hte common geometric operations(truncation, dual, augmentation, dimensishment, etc.) could be applied to an arbitrary seed, and what a 3-d analog of the flower of life pattern would look like. Sadly, I’m pretty sure a physical hyper compass would be impossible without someone inventing a real world analog of hard light holograms and if there exists any 3-d drawing program that allows for such constructions, I can’t even imagine how one could make it blind accessible in any meaningful way.
If one started with the [
Vesica Piscis construction with two spheres and added a third sphere analogous to drawing the classic 3 circle venn diagram, I think the centers of the 3 spheres and their intersection point would form the corners of a regular tetrahedron, but I’m not 100% certain of that, and I think drawing a plane through the center of a sphere and a line through the center and the circle where the sphere intersects that sphere would give you one diameter, the perpendicular bisector would give you a great circle perpendicular to the diameter that could be used to define a perpendicular diameter, and doing the bisector construction a second time on the second diameter would give you a third diameter perpendicular to the first two, and so the points to inscribe a regular octahedron in the sphere… and I thinkfinding the midpoints of the octahedron’s edges, draw the medians of the octahedron’s faces to find its face centers, and draw the lines through the center of the sphere and the center of the octahedron’s faces would project the vertices of a cube onto the sphere, but I’m at a loss for most other polyhedra… though if they work, the tetrahedron gives us the triangular bipyramid for free and the octahedron the square pyramid., so 2 down, 90 to go for Johnson solids maybe… And I think dual construction might be possible for arbitrary polyhedra, or at least for regular faced ones or ones with highly symmetric, but irregular faces(any where connecting opposite vertices, opposite edge midpoints, or vertices to an opposing edge midpoint intersect at the center of the face).
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Depends on the construction rules. The system you describe is too powerful in fact: you don’t need all those assumptions. With just lines, planes and (partial) spheres, you can construct at least all platonic and Catalan solids, I’m quite sure. Possibly others as well, but that would require tedious checking. Galois theory solves these kinds of questions: if the set of constructible points is a field, then constructing more points with additional instruments corresponds to working in a field extension. The theory is well understood if we transpose it into abstract algebra.
It’s plausible that constructing some solutions to quartics not available on planar constructions is possible by going 3D. In fact this should generalize fairly well (no reason to limit the thinking to just dimensions 2 and 3), but the precise details seem to be messy.
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