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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 - Facing NGC 3344
Nearly 40,000 light-years across, the big, beautiful spiral galaxy is located just 20 million light-years away in the constellation of Leo Minor. This multi-color Hubble Space Telescope close-up of NGC 3344 includes remarkable details from near infrared to ultraviolet wavelengths. The frame extends some 15,000 light-years across the spiral's central regions. From the core outward, the galaxy's colors change from the yellowish light of old stars in the center to young blue star clusters and reddish star forming regions along the loose, fragmented spiral arms. Of course, the bright stars with a spiky appearance are in front of NGC 3344 and lie well within our own Milky Way. APOD Turns 30!: Free Public Lecture in Anchorage on Wednesday, June 11 at 7 pm.