STREAM
My quaint space by the fireplace. Look at space, some cool paintings, listen to music, discover new books?
Artist Showcase: Charles Reid
Charles Reid has long been considered a master of the medium of watercolor. His paintings are fresh and spontaneous, displaying his profound understanding of light and color and expert drawing ability. The viewer is immediately drawn into his rich compositions. His figures are beautifully rendered, in a simple, direct and natural, gestural manner that reveals his interest in painting the light as much as the person.
Reid was born in Cambridge, New York, and studied art at the University of Vermont and the Art Students League of New York. He has won numerous awards, including the Childe Hassam Purchase Prize at the American Academy of Arts and Letters, the National Academy of Design and the American Watercolor Society. In 1980 he was elected to the National Academy of Design. Public collections of his work include Smith College, Yellowstone Art Center, Brigham Young College, Roche Corporation, and the National Academy of Design. In addition to painting, Reid teaches workshops around the world. He has written eleven books on painting in watercolor and oil. Recent awards include a Purchase Award from Shanghai International Biennial Exhibition in 2013 and a Gold Medal from the Portrait Society of America in 2013.
Theorem of the Month: Euler's Identity
Euler’s formula states that for any real number $x$:
$$ e^{ix} = \cos x + i \sin x $$This remarkable connection between the exponential function and trigonometry can be visualized by considering the power series expansions of $e^z$, $\cos x$, and $\sin x$:
$$ \begin{align*} e^z &= 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \frac{z^4}{4!} + \dots \\ \\[0.1em] \cos x &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots \\ \\[0.1pt] \sin x &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots \end{align*} $$Now, let $z = ix$. Substituting into the series for $e^z$ and grouping real and imaginary terms (remembering $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, etc.):
$$ e^{ix} = \left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots\right) + i\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots\right) $$We see that the terms in the parentheses are exactly the series for $\cos x$ and $\sin x$, respectively. To obtain Euler’s identity specifically, we set $x = \pi$. Since $\cos \pi = -1$ and $\sin \pi = 0$:
$$ e^{i\pi} = \cos \pi + i \sin \pi = -1 + i(0) = -1 $$Rearranging this gives the elegant identity:
$$ e^{i\pi} + 1 = 0 \quad \blacksquare$$Bookshelf
Space Showcase: NASA APOD - An Almost Everything Sky
First, slanting down from the upper left and far in the distance is the central band of our Milky Way Galaxy. More modestly, slanting down from the upper right and high in Earth's atmosphere is a bright meteor. The dim band of light across the central diagonal is zodiacal light: sunlight reflected from dust in the inner Solar System. The green glow on the far right is aurora high in Earth's atmosphere. The bright zigzagging bright line near the bottom is just a light that was held by the scene-planning astrophotographer. This "almost everything" sky was captured over rocks on Castle Hill, New Zealand late last month. The featured finished frame is a combination of 10 exposures all taken with the same camera and from the same location. But what about the astrophotographer himself? He's pictured too -- can you find him? Jigsaw Fun: Astronomy Puzzle of the Day.