Sol Lewitt, Complex Forms
Scutoid
The newly discovered shape that shows how nature packs cells efficiently into three-dimensional structures.
Researchers have discovered a new geometric shape that’s been hiding in plain sight.
A team studying the cells that give rise to embryos and can be found lining our organs and blood vessels pinpointed a three-dimensional shape that occurs as they bend and pack together.
The new shape, dubbed the scutoid, allows these epithelial cells to organize with the most efficiency, as opposed to the column or bottle-like shapes scientists previously attributed to this process.
Mathematics is one of the essential emanations of the human spirit—a thing to be valued in and for itself like art or poetry.
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Oswald Veblen, mathematician known for proving the Jordan curve theorem (via mathblab) |
How can you visualize a 4-dimensional graph? Real valued functions of one variable are nicely graphed on a 2D gird (1 number line for the inputs crossed with 1 number line for the outputs) and we can talk about things like slope, area under the curve, etc. But complex functions of one variable would require a 4D grid (2 number lines for the inputs crossed with 2 number lines for the outputs). Although there are some clever ways to visualize complex functions with contour plots, 3D colored plots, two 2D plots, etc., notions like the integral as the area under the curve do not cross over. Unlike with integrals learned in introductory calculus, the complex integral has no geometric realization. Consequently, complex functions are often dealt with completely abstractly.
BUT WHY BOTHER WITH 4D FUNCTIONS? 4D STUFF ISN’T EVEN REAL! Right?
For mathematics a 4th dimension is introduced whenever it is useful or of interest. Some mathematics involves arbitrary whole number dimensions and some even fractional dimensions. Often, a higher dimensional framework is useful in solving problems. For example, finding roots of polynomials sometimes requires the use of complex numbers. Also the famous sphere packing problem was solved in higher dimensions first which later gave insights into how to solve the problem in lower dimensions https://www.quantamagazine.org/20160330-sphere-packing-solved-in-higher-dimensions/. But there are more concrete examples too. Well known for this is Einstein’s work.
Oh right, I heard the 4th dimension is time!
Yes, in his theory of relativity, Einstein utilized a 4-dimensional geometry in which he assigned 3 dimensions to represent space (left-right, back-forth, up-down) and then a 4th dimension to represent time. This ended up being useful for describing the universe at a large scale.
But is there a real life 4th dimension? You know, like, is there a strange place I can go, become invisible, find monsters, travel through time, etc?
That would be cool. String theory employs high dimensional mathematics in an effort to provide a unifying framework of our physical laws. In fact, one prevailing string theory holds that there is a 4th and even more, actually 10 dimensions of space and 1 of time, but unfortunately the extra 7 spatial dimensions are so small not even sub-atomic particles can fit through, least of all you and me.
