From Scientific American:
The eight dimensions of the octonions aren’t the only interesting thing about the number eight, however. Baez highlights the number eight as one of his three favorite numbers. (The other two? Five and 24.) In 2008 Baez gave a series of lectures explaining what makes five, eight and 24 such unique and mysterious entities. The lectures, which are intended for a general interest audience, live on the Internet as both pdfs of the slides he used and video recordings. Watching them, you can learn not only a lot more about what makes octonions special, but also sphere stacking, the golden ratio, Islamic tiles, and why the sum of all integers equals –1/12.