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September 28, 2015

First space probes to visit bodies of the Solar System

Filed under: science — ckrao @ 11:11 am

The table below shows the first space probes to visit various bodies of the Solar System (and the year) by mission type (flyby, orbit, impact or soft landing). Notably 2015 had three events: MESSENGER ended its four year orbit of Mercury with the first impact on the planet, Dawn became the first spacecraft to orbit a dwarf planet (Ceres, in the asteroid belt), and New Horizons flew by Pluto.

 flyby orbit impact soft landing Sun Helios 2 1976 (within 43m km) Luna 1 1959 Mercury Mariner 10 1974 MESSENGER 2011 MESSENGER 2015 Venus Venera 1 1961 Venera 9 1975 Venera 3 1966 Venera 9 1975 Mars Mariner 4 1965 Mariner 9 1971 Mars 2 1971 Mars 3 1971 Jupiter Pioneer 10 1973 Galileo 1995 Galileo 1995 Saturn Pioneer 11 1979 Cassini 2004 Uranus Voyager 2 1986 Neptune Voyager 2 1989 Pluto New Horizons 2015 Ceres Dawn 2015 Moon Luna 1 1959 Luna 10 1966 Luna 2 1959 Luna 9 1966 Titan Huygens 2005 asteroid Galileo asteroid 951 Gaspra – 1991 NEAR Shoemaker asteroid 433 Eros – 2000 NEAR Shoemaker asteroid 433 Eros – 2000 comet ICE comet Giacobini-Zinner – 1985 Rosetta comet Churyumov-Gerasimenko – 2014 Deep Impact – Impactor comet Tempel -2005 Philae comet Churyumov-Gersimenko – 2014

September 20, 2015

The simplest Heronian triangles

Filed under: mathematics — ckrao @ 12:05 pm

Heronian triangles are those whose side lengths and area have integer value. Most of the basic ones are formed either by right-angled triangles of integer sides, or by two such triangles joined together. Following the proof in [1] it is not difficult to show that such triangles have side lengths proportional to $(x,y,z) = (n(m^2 + h^2), m(n^2 + h^2), (m+n)(mn-h^2))$ where $m,n$ and $h$ are integers with $mn > h^2$.  Firstly, if a triangle has integer side lengths and area, its altitudes must be rational, being twice the area divided by a side length. Also by the cosine rule, the cosine of its angles must be rational, so $z_1$ and $z_2$ in the diagram below are rational too (here assume $z$ is the longest side, so that the altitude is inside the triangle).

This gives us the equations

$\displaystyle h^2 = x^2 - z_1^2 = y^2 - z_2^2, z_1 + z_2 = z,\quad \quad (1)$

where $h, z_1, z_2 \in \mathbb{Q}$. Letting $x + z_1 = m$ and $y + z_2 = n$ it follows from the above equations that $x - z_1 = h^2/m, y-z^2 = h^2/n$ from which

$\displaystyle (x,y,z) = \left(\left(\frac{1}{2}(m + \frac{h^2}{m}\right), \frac{1}{2}\left(n + \frac{h^2}{n}\right), \frac{1}{2}\left( m - \frac{h^2}{m} + (n - \frac{h^2}{n}\right)\right). \quad\quad (2)$

Scaling the sides up by a factor of $2mn$, the sides are proportional to

$(x',y',z') = (n(m^2 + h^2), m(n^2 + h^2), (m+n)(mn-h^2)).\quad\quad(3)$

Next, letting $d$ be the common denominator of the rational numbers $h, z_1$ and $z_2$, we multiply the rational solution $(x', y', z')$ in (3) each by $d^3$ to obtain an integral solution. The altitude upon side length $z$ is proportional to $2hmn$ and the area is $hmn(m+n)(mn-h^2)$. Hence if we start with positive $m,n,h$ with no common factor and with $mn > h^2$, then (3) gives the side lengths of a Heronian triangle that can then be made primitive by dividing by a common factor.

Below the 20 primitive Heronian triangles with area less than 100 are illustrated to scale, where the first row has been doubled in size for easier viewing (a larger list is here). Note that all but one of them is either an integer right-angled triangle or decomposable into two such triangles as indicated by the blue numbers and sides. Refer to [2] for more on triangles which are not decomposable into two integer right-angled triangles. Here are the primitive Pythagorean triples that feature in the triangles:

• 3-4-5
• 5-12-13
• 8-15-17
• 20-21-29
• 7-24-25
• 28-45-53

References

[1] Carmichael, R. D., 1914, “Diophantine Analysis”, pp.11-13; in R. D. Carmichael, 1959, The Theory of Numbers and Diophantine Analysis, Dover Publications, Inc.

[2] Yiu, Paul (2008), Heron triangles which cannot be decomposed into two integer right triangles (PDF), 41st Meeting of Florida Section of Mathematical Association of America.

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