Most creative way of dividing a rectangle in half?

Question

Consider the rectangle ((0,0),(2,1))

What is the most creative way of dividing its area in half?
- e.g. by any combination of straight lines, triangles, curves, fractal curves (e.g. Cantor dust), star, ellipse with non-integer exponents, two concentric nautiluses, cyclic curve, other plane curve...
- doesn't have to be just two subshapes, you can use more
- e.g. you could figure out what position and angle another rectangle of equal area overlapping it at some angle would have to be to to cover half its area
- prize is for the most creative!
- also good if you can give coordinates or equations

For inspiration:
http://mathworld.wolfram.com/topics/PlaneCurves.html


(If you think this is easy, the follow-up would be "n different distinct methods of marking out collections of subshapes, each having total relative net area 1/n")

Answer 1

Have you considered

microprocessor photo-masks

rescaled stellar spectra

counts in base n numbers

text fonts

literary works in original language, translated, and transliterated

randomly generated mazes

topographical maps

scaled stock-market price & volume graphs



All of these are doable and calculable since each is a finite structure, but some of the wilder ones would involve a great deal of tedium.

edit:

Oops, forgot gerrymandering!

Answer 2

Though you have not derived de novo a new formula, your approach seems novel enough. Now, if you at the time of your discovery knew both the formula of Gauss, viz. 1+2+3+...+n = n(n+1)/2, and the formula A = bh/2 for the area of a triangle, then your unconscious mind might have made the connection based on the similarity of the formulae, phrasing it in the triangle construction you have related, particularly if you had some familiarity with Pascal's triangle. So you see, the question of independence has more to do with your prior experience and exposure to these key elements, but clearly your mind was taking these things and reconfiguring them, both literally and figuratively, in a novel and intriguing manner and so perhaps the issue of independence is a moot one. It IS a work of cleverness that reveals the workings of a keen mind! Incidentally, the sum of squares (n^2) is n(n+1)(2n+1)/6, that of cubes (n^3) is [n(n+1)/2]^2 (i.e. the sum of the cubes of integers from 1 to n is equal to the square of the sum of integers from 1 to n), etc. There are similar formulas for the sum of any power and a general formula based on the Bernoulli numbers. Have you ever considered that your passion for precisely rhymed and metered poetry is reflective of a mind that finds pattern and meaning from such associations? You see, I am a poet of similar persuasion and also a mathematician by training and, more significantly, by nature. When I was the age you are now (16) I "discovered" the closed form for the Fibonacci numbers and thought that I had made an original contribution to mathematics. I was wrong. But the point is that in the process of that "discovery" I felt actualized in a way I never had before, and that was -- is -- magical. Continue studying mathematics and writing poetry both and the dream of the Renaissance will be realized in you. Wonderful question, young man.

Answer 3

stare at it

theres always a >0 chance that it will turn into an interestoing shape exactly half of the area so therefore it must happen at some point

Answer 4

draw a sketch inside it

Answer 5

give it to our leaders of the world

Answer 6

i would cut it with scissors