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 … Continue reading The Cosmology of the Divine Comedy
Tag: Astronomy
Mercury’s Mercurial Orbit
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 … Continue reading Mercury’s Mercurial Orbit
From Sub-atomic to Cosmic Strings
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. A longstanding goal of physics is to construct a new theory … Continue reading From Sub-atomic to Cosmic Strings
Image Processing Emerges from the Shadows
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 … Continue reading Image Processing Emerges from the Shadows
Where is the Sun?
The position of the Sun in the sky depends on where we are and on the time of day. Due to the Earth's rotation, the Sun appears to cross the celestial sphere each day along a path called the ecliptic. The observer's position on Earth is given by the geographic latitude and longitude. The path … Continue reading Where is the Sun?
Gauss Predicts the Orbit of Ceres
On the first day of a new century, January 1, 1801, astronomer Giuseppe Piazzi discovered a new celestial object, the minor planet Ceres. He made about 20 observations from his observatory in Palermo before the object was lost in the glare of the Sun in early February. Later in the year, several astronomers tried without … Continue reading Gauss Predicts the Orbit of Ceres
From Impossible Shapes to the Nobel Prize
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 … Continue reading From Impossible Shapes to the Nobel Prize
Is There Anyone Out There? The Drake Equation gives a Clue
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 … Continue reading Is There Anyone Out There? The Drake Equation gives a Clue
Boxes and Loops
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. Libration and Rotation of a Pendulum The simple pendulum, with one degree of freedom, provides a … Continue reading Boxes and Loops
Kepler’s Vanishing Circles Hidden in Hamilton’s Hodograph
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 … Continue reading Kepler’s Vanishing Circles Hidden in Hamilton’s Hodograph
K3 implies the Inverse Square Law.
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 … Continue reading K3 implies the Inverse Square Law.
Trappist-1 & the Age of Aquarius
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 … Continue reading Trappist-1 & the Age of Aquarius
The 3 : 2 Resonance between Neptune and Pluto
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. The Harmony of the Spheres Pythagoras based his musical analysis on two ratios: the octave 2 : 1 and the perfect fifth 3 : 2. He … Continue reading The 3 : 2 Resonance between Neptune and Pluto
Gravitational Waves & Ringing Teacups
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. Energy and angular momentum are carried … Continue reading Gravitational Waves & Ringing Teacups
Johannes Kepler and the Song of the Earth
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 … Continue reading Johannes Kepler and the Song of the Earth
Galileo’s Book of Nature
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 … Continue reading Galileo’s Book of Nature
Darker Mornings, Brighter Evenings
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 … Continue reading Darker Mornings, Brighter Evenings
The Star of Bethlehem … or was it a Planet?
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 … Continue reading The Star of Bethlehem … or was it a Planet?
Slingshot Orbit to Asteroid Bennu
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 … Continue reading Slingshot Orbit to Asteroid Bennu
Pedro Nunes and Solar Retrogression
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 … Continue reading Pedro Nunes and Solar Retrogression
Quadrivium: The Noble Fourfold Way
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 6th century AD [see TM119 or search … Continue reading Quadrivium: The Noble Fourfold Way
A New Window on the World
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. … Continue reading A New Window on the World
Kepler’s Magnificent Mysterium Cosmographicum
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 … Continue reading Kepler’s Magnificent Mysterium Cosmographicum
The Search is on for Planet Nine
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 … Continue reading The Search is on for Planet Nine
Thomas Harriot: Mathematician, Astronomer and Navigator
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. Raleigh needed an excellent navigator on his transatlantic voyages, and he brought … Continue reading Thomas Harriot: Mathematician, Astronomer and Navigator
The Great American Eclipse
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 ]. Solar eclipses are not especially rare, but this one is of … Continue reading The Great American Eclipse
Pluto’s Unruly Family
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 … Continue reading Pluto’s Unruly Family
The Hodograph
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 idea of a hodograph is very simple. … Continue reading The Hodograph
Falling Bodies [2]: Philae
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). Rosetta was in orbit around the comet and, after detatchment, the lander … Continue reading Falling Bodies [2]: Philae
Light Weight (*)
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 … Continue reading Light Weight (*)
When did Hammurabi reign?
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 … Continue reading When did Hammurabi reign?
Solar System Perturbations
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. The Sun is one million times more … Continue reading Solar System Perturbations
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