If you think poetry and maths are poles apart, think again. Around the sixth century, Indian poet and mathematician Virahanka codified the structure of Sanskrit poetry, formulating rules for the patterns of long and short syllables. In this process, a sequence emerged in which each term is the sum of the preceding two. This is precisely the sequence studied centuries later by Leonardo Bonacci of Pisa, which we now call the Fibonacci sequence [TM241 or search for “thatsmaths” at irishtimes.com].

## Posts Tagged 'Astronomy'

### The Cosmology of the Divine Comedy

Published January 19, 2023 Irish Times Leave a CommentTags: Astronomy, Cosmology, Topology

### Mercury’s Mercurial Orbit

Published November 10, 2022 Occasional ClosedTags: Astronomy, Mechanics, Relativity

The tiny deviation of the orbit of Mercury from a pure ellipse might seem to be of no consequence. Yet the minute precession of this planet was one of the factors leading to a revolution in our world view. Attempts to explain the anomaly in the context of Newtonian mechanics were unsatisfactory. It was only with the emergence of general relativity that we were able to understand the observed phenomenon. Continue reading ‘Mercury’s Mercurial Orbit’

### From Sub-atomic to Cosmic Strings

Published October 20, 2022 Irish Times , Occasional ClosedTags: Astronomy, Physics

The two great pillars of modern physics are quantum mechanics and general relativity. These theories describe small-scale and large-scale phenomena, respectively. While quantum mechanics predicts the shape of a hydrogen atom, general relativity explains the properties of the visible universe on the largest scales.

### Image Processing Emerges from the Shadows

Published May 19, 2022 Irish Times ClosedTags: Astronomy, Trigonometry

Satellite images are of enormous importance in military contexts. A battery of mathematical and image-processing techniques allows us to extract information that can play a critical role in tactical planning and operations. The information in an image may not be immediately evident. For example, an overhead image gives no direct information about the height of buildings or industrial installations, but shadows, together with the time, date and basic trigonometry, enable heights to be determined [TM233 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Image Processing Emerges from the Shadows’

### Where is the Sun?

Published February 10, 2022 Occasional ClosedTags: Astronomy, Spherical Trigonometry

*ecliptic*. The observer’s position on Earth is given by the geographic latitude and longitude. The path of the Sun depends on the latitude and the date, while the time when the Sun crosses the local meridian is determined by the longitude.

### From Impossible Shapes to the Nobel Prize

Published October 8, 2020 Occasional ClosedTags: Astronomy, Geometry

Roger Penrose, British mathematical physicist, mathematician and philosopher of science has just been named as one of the winners of the 2020 Nobel Prize in Physics. Penrose has made major contributions to general relativity and cosmology.

Penrose has also come up with some ingenious mathematical inventions. He discovered a way of defining a pseudo-inverse for matrices that are singular, he rediscovered an “impossible object”, now called the Penrose Triangle, and he discovered that the plane could be tiled in a non-periodic way using two simple polygonal shapes called kites and darts.Continue reading ‘From Impossible Shapes to the Nobel Prize’

### Is There Anyone Out There? The Drake Equation gives a Clue

Published August 20, 2020 Irish Times ClosedTags: Astronomy

The Drake Equation is a formula for the number of developed civilizations in our galaxy, the Milky Way. This number is determined by seven factors. Some are known with good accuracy but the values of most are quite uncertain. It is a simple equation comprising seven terms multiplied together [TM193 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Is There Anyone Out There? The Drake Equation gives a Clue’

We will describe some generic behaviour patterns of dynamical systems. In many systems, the orbits exhibit characteristic patterns called boxes and loops. We first describe orbits for a simple pendulum, and then look at some systems in higher dimensions.

### Kepler’s Vanishing Circles Hidden in Hamilton’s Hodograph

Published May 2, 2019 Irish Times ClosedTags: Astronomy, Hamilton, Mechanics

The Greeks regarded the heavens as the epitome of perfection. All flaws and blemishes were confined to the terrestrial domain. Since the circle is perfect in its infinite symmetry, it was concluded by Aristotle that the Sun and planets move in circles around the Earth. Later, the astronomer Ptolemy accounted for deviations by means of additional circles, or epicycles. He stuck with the circular model [TM162 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Kepler’s Vanishing Circles Hidden in Hamilton’s Hodograph’

### K3 implies the Inverse Square Law.

Published April 25, 2019 Occasional ClosedTags: Astronomy, Mechanics

Kepler formulated three remarkable laws of planetary motion. He deduced them directly from observations of the planets, most particularly of the motion of Mars. The first two laws appeared in 1609 in Kepler’s *Astronomia Nova*. The first law (**K1**) describes the orbit of a planet as an ellipse with the Sun at one focus. The second law (**K2**) states that the radial line from Sun to planet sweeps out equal areas in equal times; we now describe this in terms of conservation of angular momentum.

The third law (**K3**), which appeared in 1619 in Kepler’s *Harmonices Mundi*, is of a different character. It does not relate to a single planet, but connects the motions of different planets. It states that the squares of the orbital periods vary in proportion to the cubes of the semi-major axes. For circular orbits, the period squared is proportional to the radius cubed.

### Trappist-1 & the Age of Aquarius

Published January 3, 2019 Irish Times ClosedTags: Astronomy, Mechanics

The Pythagoreans believed that the planets generate sounds as they move through the cosmos. The idea of the harmony of the spheres was brought to a high level by Johannes Kepler in his book *Harmonices Mundi*, where he identified many simple relationships between the orbital periods of the planets [TM154 or search for “thatsmaths” at irishtimes.com].

Kepler’s idea was not much supported by his contemporaries, but in recent times astronomers have come to realize that resonances amongst the orbits has a crucial dynamical function. Continue reading ‘Trappist-1 & the Age of Aquarius’

### The 3 : 2 Resonance between Neptune and Pluto

Published December 13, 2018 Occasional ClosedTags: Astronomy

For every two orbits of Pluto around the Sun, Neptune completes three orbits. This 3 : 2 resonance has profound consequences for the stability of the orbit of Pluto.

Continue reading ‘The 3 : 2 Resonance between Neptune and Pluto’

### Gravitational Waves & Ringing Teacups

Published November 22, 2018 Occasional ClosedTags: Astronomy, Relativity, Wave Motion

Newton’s law of gravitation describes how two celestial bodies orbit one another, each tracing out an elliptical path. But this is imprecise: the theory of general relativity shows that two such bodies radiate energy away in the form of * gravitational waves* (GWs), and spiral inwards until they eventually collide.

### Johannes Kepler and the Song of the Earth

Published November 1, 2018 Irish Times ClosedTags: Astronomy, Music

Johannes Kepler, German mathematician and astronomer, sought to explain the solar system in terms of divine harmony. His goal was to find a system of the world that was mathematically correct and harmonically pleasing. His methodology was scientific in that his hypotheses were inspired by and confirmed by observations. However, his theological training and astrological interests influenced his thinking [TM150, or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Johannes Kepler and the Song of the Earth’

### Galileo’s Book of Nature

Published February 15, 2018 Irish Times ClosedTags: Astronomy, History, Mechanics

In 1971, astronaut David Scott, standing on the Moon, dropped a hammer and a feather and found that both reached the surface at the same time. This popular experiment during the Apollo 15 mission was a dramatic demonstration of a prediction made by Galileo three centuries earlier. Galileo was born in Pisa on 15 February 1564, just 454 years ago today [TM133 or search for “thatsmaths” at irishtimes.com].

### Darker Mornings, Brighter Evenings

Published December 21, 2017 Irish Times ClosedTags: Astronomy, Time measurement

Today is the winter solstice, the shortest day of the year. We might expect that the latest sunrise and earliest sunset also occur today. In fact, the earliest sunset, the darkest day of the year, was on 13 December, over a week ago, and the latest sunrise is still more than a week away. This curious behaviour is due to the unsteady path of the Earth around the Sun. Our clocks, which run regularly at what is called mean time, move in and out of synchronization with solar time [TM129 or search for “thatsmaths” at irishtimes.com].

### The Star of Bethlehem … or was it a Planet?

Published December 7, 2017 Irish Times ClosedTags: Astronomy

People of old were more aware than we are of the night sky and took a keen interest in unusual happenings above them. The configuration of the stars was believed to be linked to human affairs and many astronomical phenomena were interpreted as signs of good or evil in the offing. The Three Wise Men or Magi were astrologers, experts in celestial matters, and would have drawn inferences from what they observed in the sky [TM128 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘The Star of Bethlehem … or was it a Planet?’

### Slingshot Orbit to Asteroid Bennu

Published November 16, 2017 Irish Times ClosedTags: Astronomy, Mechanics

The *Voyager 1* and *Voyager 2* spacecraft have now left the solar system and will continue into deep space. How did we manage to send them so far? The *Voyager* spacecraft used gravity assists to visit Jupiter, Saturn, Uranus and Neptune in the late 1970s and 1980s. Gravity assist manoeuvres, known as **slingshots**, are essential for interplanetary missions. They were first used in the Soviet Luna-3 mission in 1959, when images of the far side of the Moon were obtained. Space mission planners use them because they require no fuel and the gain in speed dramatically shortens the time of missions to the outer planets.

### Pedro Nunes and Solar Retrogression

Published October 12, 2017 Occasional ClosedTags: Astronomy, Time measurement

In northern latitudes we are used to the Sun rising in the East, following a smooth and even course through the southern sky and setting in the West. The idea that the compass bearing of the Sun might reverse seems fanciful. But in 1537 Portuguese mathematician Pedro Nunes showed that the shadow cast by the gnomon of a sun dial can move backwards.

Nunes’ prediction was counter-intuitive. It came long before Newton, Galileo and Kepler, and Copernicus’ heliocentric theory had not yet been published. The retrogression was a remarkable example of the power of mathematics to predict physical behaviour.

### Quadrivium: The Noble Fourfold Way

Published July 20, 2017 Irish Times ClosedTags: Arithmetic, Astronomy, Geometry, History, Music

According to Plato, a core of mathematical knowledge – later known as the Quadrivium – was essential for an understanding of the Universe. The curriculum was outlined in Plato’s *Republic*. The name Quadrivium means four ways, but this term was not used until the time of Boethius in the 6^{th} century AD [see TM119 or search for “thatsmaths” at irishtimes.com].

It is said that an inscription over the entrance to Plato’s Academy read “Let None But Geometers Enter Here”. This indicated that the Quadrivium was a prerequisite for the study of philosophy in ancient Greece.

The motto of the Pythagoreans was “All is Number” and Pythagoras may have been the first person to imagine that the workings of the world might be understood in mathematical terms. This idea has now brought us to the point where, at a fundamental level, mathematics is the primary means of describing the physical world. Galileo put it this way: the book of nature is written in the language of mathematics [TM102, or search for “thatsmaths” at irishtimes.com].

### Kepler’s Magnificent Mysterium Cosmographicum

Published October 13, 2016 Occasional ClosedTags: Astronomy, Geometry, History

Johannes Kepler’s amazing book, *Mysterium Cosmographicum*, was published in 1596. Kepler’s central idea was that the distance relationships between the six planets (only six were known at that time) could be represented by six spheres separated by the five Platonic solids. For each of these regular polyhedra, there is an inner and an outer sphere. The inner sphere is tangent to the centre of each face and the outer sphere contains all the vertices of the polyhedron.

Continue reading ‘Kepler’s Magnificent Mysterium Cosmographicum’

Then felt I like some watcher of the skies

When a new planet swims into his ken.

John Keats: *On First Looking into Chapman’s Homer*

Pluto, discovered in 1930, orbits in the Kuiper Belt, a ring of asteroids and icy debris beyond Neptune. About ten years ago, it was reclassified as a “dwarf planet”, so now we have eight planets. Is there a ninth anywhere? A recent paper in the *Astronomical Journal* presents evidence of a large planet in an orbit far beyond Neptune. If confirmed, this would be a dramatic confirmation of our understanding of solar system dynamics [TM088, or search for “thatsmaths” at irishtimes.com].

### Thomas Harriot: Mathematician, Astronomer and Navigator

Published August 27, 2015 Occasional ClosedTags: Astronomy, History

Sir Walter Raleigh, adventurer, explorer and privateer, was among most colourful characters of Tudor times. He acquired extensive estates in Waterford and Cork, including Molana Abbey near Youghal, which he gave to his friend and advisor, the brilliant mathematician and astronomer Thomas Harriot.

Continue reading ‘Thomas Harriot: Mathematician, Astronomer and Navigator’

### The Great American Eclipse

Published August 20, 2015 Irish Times ClosedTags: Astronomy, Time measurement

Just two years from now, on Monday, August 21, 2017, the Moon’s shadow will sweep across the United States at a speed of over 2,000 km/hr. The Great American Eclipse of 2017 will generate a frenzy of activity. [TM074: search for “thatsmaths” at irishtimes.com ].

An astrodynamical miracle is happening in the sky above. Our ability to launch a space-probe from the revolving Earth to reach a moving target billions of kilometres away almost ten years later with pinpoint accuracy is truly astounding. “New Horizons” promises to enhance our knowledge of the solar system and it may help us to understand our own planet too.

[See this week’s That’s Maths column (TM071): search for “thatsmaths” at irishtimes.com].The Hodograph is a vector diagram showing how velocity changes with position or time. It was made popular by William Rowan Hamilton who, in 1847, gave an account of it in the Proceedings of the Royal Irish Academy. Hodographs are valuable in fluid dynamics, astronomy and meteorology.

The **ESA Rosetta Mission**, launched in March 2004, rendezvoused with comet 67P/C-G in August 2014. The lander Philae touched down on the comet on 12 November and came to rest after bouncing twice (the harpoon tethers and cold gas retro-jet failed to fire).

### Light Weight (*)

Published October 30, 2014 Occasional ClosedTags: Applied Maths, Astronomy, Mechanics, Physics

Does light have weight? Newton thought that light was influenced by gravity and, using his laws of motion, we can calculate how gravity bends a light beam. The effect is observable during a total eclipse of the sun: photographs of the sky are compared with the same region when the sun is elsewhere and a radial displacement of the star images is found. But the amount predicted by Newton’s laws is only half the observed value.

### When did Hammurabi reign?

Published June 19, 2014 Irish Times ClosedTags: Astronomy, History, Time measurement

The consequences of the Earth’s changing climate may be very grave. It is essential to understand past climate change so that we can anticipate future changes. This week, *That’s Maths* in *The Irish Times* ( TM047 ) is about the chronology of the Middle East. Surprisingly, this has important implications for our understanding of climate change.

Remarkable progress in understanding the dynamics of the planets has been possible thanks to their relatively small masses and the overwhelming dominance of the Sun. The figure below shows the relative masses of the Sun, planets and some natural satellites, taking the mass of Earth to be unity.