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Yes, that’s North America on Jupiter.

Yes, that’s North America on Jupiter.

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The plastic number

The sequence of side lengths of equilateral triangles in this picture form the Padovan sequence (1,1,1,2,2,3,4,5,7,9…). Just as the Fibonacci sequence is governed by the properties of the golden ratio, the Padovan sequence is governed by the properties of the so-called plastic number.

The Padovan sequence P(n) is sequence is defined by setting P(1)=P(2)=P(3)=1, and then requiring P(n) = P(n–2) + P(n–3) for n > 3. The generating function for the sequence is given by G(x)=(1+x)/(1–x^2–x^3), which means that if this ratio of polynomials is expanded as a power series in x, the coefficient in G(x) of x^n (i.e., x to the nth power) is equal to P(n). 

The denominator in the formula for the generating function, 1–x^2–x^3, can be regarded as an algebraic encoding of the recurrence relation P(n)–P(n–2)–P(n–3)=0. An easy calculation involving polynomials shows that the product (1–x^2–x^3)(1–x+x^2) is given by 1–x–x^5. This means that the generating function G(x) can be rewritten so that the denominator polynomial is given by 1–x–x^5, which in turn means that, if n is large enough, the sequence will satisfy the recurrence relation P(n) = P(n–1) + P(n–5). 

The picture is an illustration of this last relation. Notice that the big triangle with side 16 is bounded on one side by the preceding triangle in the sequence (of side 12) and the triangle five places earlier in the sequence (of side 4). This corresponds to the fact that P(n) = P(n–1) + P(n–5) for n=12 and P(n)=16.

It turns out that the polynomial 1–x^2–x^3 has exactly one real root, and the plastic number is the reciprocal of this root. Another way to say this is that the plastic number is the unique real solution of the equation x^3=x+1. It is not hard to show using abstract algebra that this solution is an irrational number; the same is true for the golden ratio, which is the larger real solution of the equation x^2=x+1. The decimal expansion of the plastic number is therefore non-recurring; the first few digits are 1.324717957…, and over 10,000,000,000 digits have been computed. 

The plastic number is mathematically significant because it is the smallest Pisot number. A Pisot number is a real root of a monic integer polynomial whose other roots are complex numbers of absolute value less than 1. The word monic means that the highest power of x occurring has a coefficient of 1. The connection with the Padovan sequence is that the ratio P(n+1)/P(n), as n becomes large, tends to the plastic number. (In the case of the Fibonacci numbers, the corresponding ratio approaches the golden ratio.)

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Astonishing image of planet-forming disk from ALMA
http://bit.ly/1x6G691

It looks like an artist’s illustration, but it’s a real image and it lets us glimpse the process by which planets are born in orbit around their stars.

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Called the Tree of 40 Fruit, this tree produces an array of stone fruit varieties including plums, peaches, apricots, nectarines, cherries and almonds, every year. Sixteen of these trees are now growing in the US.

Read more at http://bit.ly/1nMSSzb

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Matando Aula

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Plitvice Lakes National Park, Croatia.
Within the boundaries of this heavily forested national park, 16 crystalline lakes tumble into each other via a series of waterfalls and cascades. The mineral-rich waters carve through the rock, depositing tufa in continually changing formations. Clouds of butterflies drift above the 18km of wooden footbridges and pathways which snake around the edges and under and across the rumbling water.

It takes upwards of six hours to explore the lakes on foot, or you can slice two hours off by taking advantage of the park’s free boats and buses (departing every 30 minutes from April to October). From Entrance 2, catch the bus to the top of the upper lakes and wander back down to the shore of Kozjak , the park’s largest lake (about 4km in length). A boat will whisk you from here to the lower lakes, where the circuit culminates in the aptly named Veliki Slap , the tallest waterfall in Croatia (78m). The path then climbs steeply (offering great views and photo opportunities) to a bus stop, where you can grab a lift back to Entrance 2.

If you’ve got limited time, the upper lake section can be completed in two hours. The lower section takes about three, although we recommend that you start with the bus ride and end with the boat to save yourself a climb.

Rowboats can be hired from the shores of Lake Kozjak near Entrance 2 (50KN per hour). Note: swimming is not permitted in any of the lakes.

Source: www.lonelyplanet.com

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