![]() He tried to capitalize on his discovery by constructing a clock whose pendulum was constrained to swing between two adjacent arcs of a cycloid, so that the period of oscillations would be independent of the amplitude. To his surprise, Huygens found that the curve is an arc of an inverted cycloid. This problem is known as the tautochrone (from the Greek words meaning “the same time”). In 1673, the Dutch physicist Christiaan Huygens (1629–1695) solved one of the outstanding problems that had intrigued 17th-century scientists: to find the curve down which a particle, moving only under the force of gravity, will take the same amount of time to reach a given final point, regardless of the initial position of the particle. (In a linear spiral the distance from the center increases arithmetically-that is, in equal amounts-as in the grooves of a vinyl record.) The linear spiral on Bernoulli’s headstone can still be seen at the cloisters of the Basel Münster, perched high on a steep hill overlooking the Rhine River. His wish was fulfilled, though not quite as he had intended: For some reason, the mason engraved a linear spiral instead of a logarithmic one. He was so enamored with the logarithmic spiral that he dubbed it “spira mirabilis” and ordered it to be engraved on his tombstone after his death. Bernoulli was the senior member of an eminent dynasty of mathematicians hailing from the town of Basel. The logarithmic spiral has been known since ancient times, but it was the Swiss mathematician Jakob Bernoulli (1654–1705) who discovered most of its properties. The snail residing inside the shell gradually relocates from one chamber to the next, slightly larger chamber, yet all chambers are exactly similar to one another: A single blueprint serves them all. ![]() This property is manifested beautifully in the nautilus shell ( left). As a consequence, any sector with given angular width Δθ is similar to any other sector with the same angular width, regardless of how large or small it is. For this reason, the curve is also known as an equiangular spiral. For example, a straight line from the pole O to any point on the spiral intercepts it at a constant angle α. The many intriguing aspects of the logarithmic spiral all derive from this single feature.
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