Latest news with #Euclidean
The Hindu
26-05-2025
- General
- The Hindu
The maths of how India's coastline lengthened without gaining new land
In December 2024, the Union Ministry of Home Affairs made an important announcement as part of its 2023-2024 annual report. It said the length of India's coastline had increased from 7,516.6 km to 11,098.8 km, and that the length is also currently under review. The 7,516.6 km figure was first recorded in the 1970s based on measurement techniques available at the time. The new revised figure wasn't prompted by any territorial expansion through new land/island annexation or geological upheaval, like tectonic activity stretching the shores. The last coastal State to join the Union of India was Goa in 1961 and the only other State that joined after — Sikkim in 1975 — is landlocked. The enclaves India exchanged with Bangladesh in 2015 also lie deep inland. So what changed? The root of the discrepancy lies in geometry, in a problem called the coastline paradox. The previous estimate from the 1970s banked on maps that displayed India's coastline at a 1:4,500,000 resolution, which is too coarse to capture intricate features like estuaries, tidal creeks, sandbars, and coastal ridges. Many island groups, particularly the Andaman & Nicobar and Lakshadweep, also hadn't been comprehensively mapped or included. The more recent updated measurement — performed by the National Hydrographic Office (NHO) and the Survey of India — used electronic navigation charts at a much finer scale of 1:250,000. Preparing these charts requires the use of technologies like geographic information systems, satellite altimetry, LIDAR-GPS, and drone-based imaging. The government has also said the coastline length will be revised every 10 years from 2024-2025, as per the report. The Survey of India used highwater lines prepared by the NHO based on 2011 data on electronic navigation charts to measure the coastline. The highwater line was used as the base reference and river mouths and creeks were closed off at a fixed threshold inland. The review also included islands exposed to low tide. But for all these advances, there is a limitation — and that comes from geometry. The coastline as puzzle What's the difference between straight lines and ragged curves? In Euclidean geometry, the length of a straight line is the shortest distance between two points at the ends of the line. Curves on the other hand are measured by their geodesic length: i.e. the length along the surface of the curve. But what happens when the curve itself is irregular, jagged, and ever-changing the way a coastline is when it's shaped by river mouths, creeks, delta formations, etc.? The problem becomes harder when one attempts to draw a boundary at a river mouth: should it be marked at the ocean opening or traced further inland? Such ambiguities add to the complexity together with constant tidal fluctuations and shifting sedimentation. This is where traditional measurement concepts break down and the choice of scale becomes decisive. The coastline paradox The British mathematician and physicist Lewis Fry Richardson first identified the coastline paradox in the early 1950s. His Polish-French peer Benoît Mandelbrot examined the problem mathematically in 1967 and also popularised it. Mandelbrot found that coastlines exhibit properties similar to fractals. In a landmark paper entitled 'How Long is the Coast of Britain?', Mandelbrot explored why the length of Britain's coast varied dramatically depending on the length of the measuring stick. Using different ruler sizes on a map, he found that Britain's coast could vary from around 2,400 km to more than 3,400 km — a striking range for a fixed landmass. Note that coastlines are not true fractals in the pure mathematical sense but display fractal-like properties. To describe fractals, scientists use the concept of fractal dimension, a number that denotes the degree of complexity a shape exhibits as one zooms into it. For example, measuring a coastline with a 200-km-long ruler would smooth over most inlets and bends — but a 50-km ruler would detect them. At 1 km, the measurement will capture every estuary, tidal flat, and creek. So the more one refines the scale of the ruler, the longer the total coast becomes. Hypothetically, using a measurement unit the size of a water molecule would result in a coastline length approaching infinity. This dependence on scale underscores the inherent paradox: a finite piece of geography yielding a seemingly infinite measurement in cartography. Implications for security, fishing The change in length is not just a mathematical curiosity or an academic pursuit. The length of India's coastline influences maritime security plans, disaster preparedness (especially for cyclones and tsunamis), and fishing rights. A longer coastline obviously means a longer length to protect but it also means a longer economic zone. India has 11 coastal States and two large island groups, faces regular cyclones, and is especially vulnerable to sea-level rise. Understanding the true extent of the national coast can thus help refine climate models, coastal zoning regulations, and disaster response strategies. In the same vein, high-school geography textbooks may need to be revised as well. The coastline paradox also reveals more than a peculiar measurement challenge: it underscores how science evolves with better tools. What once appeared to be a fixed value turns fluid when examined more closely — not because the coast moved but because our eyes sharpened. India's redefined 11,099-km coastline is a testament to this progress. C. Aravinda is an academic and public health physician. The views expressed are personal.
Scottish Sun
24-05-2025
- General
- Scottish Sun
‘Absolutely bonkers!' Dad left baffled by 10-year-old's maths homework – but can you work it out
Can you solve the puzzle that has the web bewildered HEAD SCRATCHER 'Absolutely bonkers!' Dad left baffled by 10-year-old's maths homework – but can you work it out Click to share on X/Twitter (Opens in new window) Click to share on Facebook (Opens in new window) A CONFUSED dad has been left stumped by his 10-year-old son's maths homework with the internet dubbing it 'absolutely bonkers.' The father took to Reddit after being left puzzled by a multiple choice question given to his primary school-aged child. Sign up for Scottish Sun newsletter Sign up 1 Can you work out the primary school maths problem? Credit: Reddit While there are four different 'answers' to choose from, the concerned dad noted there 'must be missing something' from the equation – as none of the options appeared to be correct. 'This particular question was in my son's math homework from the other day,' he wrote. 'They reviewed the answers in class today and apparently the answer was A.' The question that has him, and everyone else, scratching their heads was this: Kayla has 18 bottles of bubbles. She wants to give two bottles to each of her six friends. How many bottles will she have left over? Children then had the option of four expressions and were challenged to identify the one that 'solves the problem'. It read: A) (18 divide 2) divide 6, B) (18 divide 2) + 6, C) (18 x 2) – 6, or D) (18 x 2) + 6. But as the dad – who said he was 'curious how they came to this answer' – pointed out, 'none of the options seemed right as I was expecting it to be 18 – (6 × 2)'. Some people rushed to the comments section to try and work out the answer, but most agreed there was an issue with the options provided with describing it as 'absolutely bonkers'. 'I think it's more likely a typo or misprint. If they swapped the subtraction and multiplication sign and moved the parentheses on answer choice C, then: (18 x 2) – 6 could become 18 – (2 x 6),' one said. 'You're correct, the teacher is wrong. If you simplify A, you get 1.5 which doesn't make any sense in the context of the problem,' mused another. 99 percent of people can't find the hidden objects in under 20 seconds - are you up to the challenge While one simply said: 'Seems that A is wrong to me too.' Some tried to make it work, but struggled. 'Really twisting my brain here to make sense of A being correct, but here goes: if you divide 18 bottles by 2 you get 9 bottles in two separate piles. Now give one bottle from each pile to all 6 friends. The result would be 3 bottles leftover in two separate piles, or 6 leftover bottles total. Gymnastics,' declared one. 'A, if they are supposed to use Euclidean divisions (18/2 = she has 9 batches of 2, 9/6 => 1 and remainder is 3),' tried another. While one described it as 'bad logic', but gave it a stab anyway. How can optical illusions and brainteasers help me? Engaging in activities like solving optical illusions and brainteasers can have many cognitive benefits as it can stimulate various brain regions. Some benefits include: Cognitive stimulation : Engaging in these activities challenges the brain, promoting mental agility and flexibility. : Engaging in these activities challenges the brain, promoting mental agility and flexibility. Problem-solving skills : Regular practice enhances analytical thinking and problem-solving abilities. : Regular practice enhances analytical thinking and problem-solving abilities. Memory improvement: These challenges often require memory recall and can contribute to better memory function. These challenges often require memory recall and can contribute to better memory function. Creativity: They encourage thinking outside the box, fostering creativity and innovative thought processes. They encourage thinking outside the box, fostering creativity and innovative thought processes. Focus and attention: Working on optical illusions and brainteasers requires concentration, contributing to improved focus. Working on optical illusions and brainteasers requires concentration, contributing to improved focus. Stress relief: The enjoyable nature of these puzzles can act as a form of relaxation and stress relief. 'This is the only way I can get any of the answer choices (and it is A) – I'm not saying it's correct, only wanted to explain their (wrong) logic: 'She's splitting the 18 bottles into sets of 2, that's 18 / 2. Then, she's splitting those sets of 2 among her 6 friends. 'That's why you divide by 6 next. That leaves you with A. But as everyone here has said, you and your son are correct. The worksheet is wrong.' Eventually, the child decided to expose the issue with the question, writing: 'None, 18 – (6 x 2).' The father later returned to update everyone, stating 'the worksheet is indeed wrong'. 'I did talk with the teacher and they went over it in class together. The teacher mentioned none of the answers were right and what my son came up with was correct,' he shared. This article was originally published on and has been republished here with permission.
The Irish Sun
24-05-2025
- General
- The Irish Sun
‘Absolutely bonkers!' Dad left baffled by 10-year-old's maths homework – but can you work it out
A CONFUSED dad has been left stumped by his 10-year-old son's maths homework with the internet dubbing it 'absolutely bonkers.' The father took to Advertisement 1 Can you work out the primary school maths problem? Credit: Reddit While there are four different 'answers' to choose from, the concerned dad noted there 'must be missing something' from the equation – as none of the options appeared to be correct. 'This particular question was in my son's math 'They reviewed the answers in class today and apparently the answer was A.' The question that has him, and everyone else, scratching their heads was this: Kayla has 18 bottles of bubbles. She wants to give two bottles to each of her six friends. How many bottles will she have left over? Advertisement Read More on Brainteasers Children then had the option of four expressions and were challenged to identify the one that 'solves the problem'. It read: A) (18 divide 2) divide 6, B) (18 divide 2) + 6, C) (18 x 2) – 6, or D) (18 x 2) + 6. But as the dad – who said he was 'curious how they came to this answer' – pointed out, 'none of the options seemed right as I was expecting it to be 18 – (6 × 2)'. Some people rushed to the comments section to try and work out the answer, but most agreed there was an issue with the options provided with describing it as 'absolutely bonkers'. Advertisement Most read in Fabulous 'I think it's more likely a typo or misprint. If they swapped the subtraction and multiplication sign and moved the parentheses on answer choice C, then: (18 x 2) – 6 could become 18 – (2 x 6),' one said. 'You're correct, the teacher is wrong. If you simplify A, you get 1.5 which doesn't make any sense in the context of the problem,' mused another. 99 percent of people can't find the hidden objects in under 20 seconds - are you up to the challenge While one simply said: 'Seems that A is wrong to me too.' Some tried to make it work, but struggled. Advertisement 'Really twisting my brain here to make sense of A being correct, but here goes: if you divide 18 bottles by 2 you get 9 bottles in two separate piles. Now give one bottle from each pile to all 6 friends. The result would be 3 bottles leftover in two separate piles, or 6 leftover bottles total. Gymnastics,' declared one. 'A, if they are supposed to use Euclidean divisions (18/2 = she has 9 batches of 2, 9/6 => 1 and remainder is 3),' tried another. While one described it as 'bad logic', but gave it a stab anyway. How can optical illusions and brainteasers help me? Engaging in activities like solving optical illusions and brainteasers can have many cognitive benefits as it can stimulate various brain regions. Some benefits include: Cognitive stimulation : Engaging in these activities challenges the brain, promoting mental agility and flexibility. Problem-solving skills : Regular practice enhances analytical thinking and problem-solving abilities. Memory improvement: These challenges often require memory recall and can contribute to better memory function. Creativity: They encourage thinking outside the box, fostering creativity and innovative thought processes. Focus and attention: Working on optical illusions and brainteasers requires concentration, contributing to improved focus. Stress relief: The enjoyable nature of these puzzles can act as a form of relaxation and stress relief. 'This is the only way I can get any of the answer choices (and it is A) – I'm not saying it's correct, only wanted to explain their (wrong) logic: Advertisement 'She's splitting the 18 bottles into sets of 2, that's 18 / 2. Then, she's splitting those sets of 2 among her 6 friends. 'That's why you divide by 6 next. That leaves you with A. But as everyone here has said, you and your son are correct. The worksheet is wrong.' Eventually, the child decided to expose the issue with the question, writing: 'None, 18 – (6 x 2).' The father later returned to update everyone, stating 'the worksheet is indeed wrong'. Advertisement 'I did talk with the teacher and they went over it in class together. The teacher mentioned none of the answers were right and what my son came up with was correct,' he shared. This article was originally published on
New York Post
23-05-2025
- General
- New York Post
Dad stumped by 10-year-old son's math homework: ‘Must be missing something'
A confused dad has been left stumped by his 10-year-old son's math homework, so he's turned to the internet for help. The American father took to Reddit after being left puzzled by a multiple-choice question given to his primary school-aged child. While there are four different 'answers' to choose from, the concerned dad noted there 'must be missing something' from the equation, as none of the options appeared to be correct. 'This particular question was in my son's math homework from the other day,' he wrote. 'They reviewed the answers in class today, and apparently the answer was A.' 5 A confused dad has been left stumped by his 10-year-old son's math homework, so he's turned to the internet for help. deagreez – The question that has him, and everyone else, scratching their heads was this: Kayla has 18 bottles of bubbles. She wants to give two bottles to each of her six friends. How many bottles will she have left over? Kids then had the option of four expressions and were challenged to identify the one that 'solves the problem.' It read: A) (18 divide 2) divide 6, B) (18 divide 2) + 6, C) (18 x 2) – 6, or D) (18 x 2) + 6. 5 The question that has him scratching his head was this: 'Kayla has 18 bottles of bubbles. She wants to give two bottles to each of her six friends. How many bottles will she have left over?' Reddit/News AU But as the dad, who said he was 'curious how they came to this answer' – pointed out, 'none of the options seemed right as I was expecting it to be 18 – (6 × 2).' Some people rushed to the comments section to try and work out the answer, but most agreed there was an issue with the options provided. 'I think it's more likely a typo or misprint. If they swapped the subtraction and multiplication sign and moved the parentheses on answer choice C, then: (18 x 2) – 6 could become 18 – (2 x 6),' one said. 'You're correct, the teacher is wrong. If you simplify A, you get 1.5, which doesn't make any sense in the context of the problem,' mused another. 5 But as the dad, who said he was 'curious how they came to this answer' – pointed out, 'none of the options seemed right as I was expecting it to be 18 – (6 × 2).' vchalup – While one simply said, 'Seems that A is wrong to me, too.' Some tried to make it work, but struggled. 'Really twisting my brain here to make sense of A being correct, but here goes: if you divide 18 bottles by 2, you get 9 bottles in two separate piles. Now, give one bottle from each pile to all 6 friends. The result would be 3 bottles left in two separate piles, or 6 leftover bottles total. Gymnastics,' declared one. 5 Some people rushed to the comments section to try and work out the answer, but most agreed there was an issue with the options provided. Tatiana Cheremukhina – 'A, if they are supposed to use Euclidean divisions (18/2 = she has 9 batches of 2, 9/6 => 1 and remainder is 3),' tried another. While one described it as 'bad logic', but gave it a stab anyway. 'This is the only way I can get any of the answer choices (and it is A) – I'm not saying it's correct, only wanted to explain their (wrong) logic: 'She's splitting the 18 bottles into sets of 2; that's 18 / 2. Then, she's splitting those sets of 2 among her 6 friends. 5 'A, if they are supposed to use Euclidean divisions (18/2 = she has 9 batches of 2, 9/6 => 1 and remainder is 3),' tried another person in the comments. olly – 'That's why you divide by 6 next. That leaves you with A. But as everyone here has said, you and your son are correct. The worksheet is wrong.' Eventually, the child decided to expose the issue with the question, writing: 'None, 18 – (6 x 2).' The father later returned to update everyone, stating, 'The worksheet is indeed wrong.' 'I did talk with the teacher, and they went over it in class together. The teacher mentioned none of the answers were right and what my son came up with was correct,' he shared.
Scientific American
28-04-2025
- Science
- Scientific American
This Cutting-Edge Encryption Originates in Renaissance Art and Math
The portly, balding sculptor-turned-architect must have drawn a few curious gazes as he set up a complicated painting apparatus in the corner of a Renaissance-era piazza. He planted his instrument, which involved an easel, a mirror and a wire framework, near the then unfinished cathedral of Florence in Italy—a cathedral whose monumental dome he would soon design. His name was Filippo Brunelleschi, and he was using the apparatus to create a painting of the baptistry near the cathedral. This demonstration of his recently discovered laws of perspective is said to have occurred sometime between 1415 and 1420, if his biographers are correct. The use of the laws of perspective amazed bystanders, altered the course of Western art for more than 450 years and, more recently, led to mathematical discoveries that enable elliptic curve cryptography. This is the security scheme that underpins Bitcoin and other cryptocurrencies and has become a fast-growing encryption method on other Internet platforms as well. But how did Renaissance art lead to the mathematics that govern modern cryptography? The tale spans six centuries and two continents and touches on infinity itself. Its characters include a French prisoner of war and two mathematicians struck down in their prime—one by illness and the other by a duelist's pistol. On supporting science journalism If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today. Merging Perspective and Geometry The first steps in the path from Brunelleschi to Bitcoin involved connecting the visual geometry within the rules for perspective to Euclidean geometry, the orderly realm of lines and points that we're taught in school. French mathematician Girard Desargues, who researched the geometry of perspective in the 17th century, was the first contributor. His findings, however, were couched in rather obscure language and struggled to find an audience. His key contributions were included in a book that had a print run of 50 copies, small even for that era, and many of those copies were eventually bought back by the publisher and destroyed. During Desargues's lifetime, only fellow French mathematician Blaise Pascal became an ardent disciple of his work. Pascal contributed his own theorem to the study of what became known as 'projective geometry.' Despite Desargues's obscurity, he made a revolutionary breakthrough by adding the concept of points and lines at infinity to Euclidean geometry. By including those points, projective geometry could be merged with Euclidean geometry in a way that was consistent for both systems. In Desargues's system, every pair of lines meets at exactly one point, with no special exceptions for parallel lines. Furthermore, parabolas and hyperbolas are equivalent to ellipses, with the addition of one or two points at infinity, respectively. These insights, though valuable, would languish in obscurity for more than 100 years. When they reappeared, it was not because Desargues's work was rediscovered. Rather, a different French mathematician, Gaspard Monge, began to work on the same questions and derived similar results. A Mathematician at War The most comprehensive work on projective geometry in this era, however, came in the 19th century from French engineer and mathematician Jean-Victor Poncelet, under rather trying circumstances. Poncelet attended France's prestigious École Polytechnique, graduating in 1810. He then joined France's corps of military engineers as a lieutenant and was ordered to what is now Belarus to support Napoleon's invasion of Russia in 1812. He and his fellow troops overran a burned out and abandoned Moscow in September of that year, and when the Russians refused to sue for peace after losing the city, Poncelet was with Napoleon when the army left Moscow and began the return to France. Poncelet remained with the French army right up to the Battle of Krasnoye in Russia, where he was separated from his unit and possibly left for dead. After the battle, he was scooped up by the Russian army and marched to Saratov, Russia, more than 700 miles from Krasnoye and more than 2,000 miles from his home in Metz, France. Although Poncelet was not confined to a prison, he was 'deprived of books and comforts of all sorts,' according to an English translation of his introduction to his first book on projective geometry. As a coping mechanism, he decided that he would try to redevelop all the math he had learned up to that point. He could not carry out this plan, however, saying that he was 'distressed above all by the misfortune of my country and my own lot.' Instead he essentially expanded on Monge's work and recreated Desargues's work independently. In hindsight, it is perhaps not surprising that a prisoner of war thousands of miles from home and unsure of when, or even if, he would be repatriated would focus his efforts on understanding points at infinity—a distance that might have seemed quite intelligible to someone in Poncelet's situation. After the war that had been sparked by that invasion ended, Poncelet returned to France and his two-volume work on projective geometry, published in 1822, was far more well-received and widely read than Desargues's work. Integrals and Curves At around the same time that Poncelet was finishing his book on projective geometry, Norwegian mathematician Niels Henrik Abel was studying elliptic integrals. These integrals are rather difficult expressions that started off as parts of an attempt to measure the circumference of an ellipse. Abel discovered that there are certain circumstances where the inverse of these elliptic integrals—what are called elliptic curves—could be used instead. The curves, it turned out, are much easier to work with. Further research into elliptic curves would be left to others, however; Abel died from tuberculosis at age 26 in 1829, mere months after publishing an important paper on the subject. In the early 1830s French mathematician Évariste Galois laid the groundwork for a new field of mathematics. Galois would die tragically but also stubbornly in a duel at age 20, but before his death he laid out the principles of group theory, in which mathematical objects and operations that follow certain rules constitute a group. The French had managed to unite projective geometry with Euclidean geometry, but it would fall to a German mathematician, August Möbius (of Möbius strip fame) to figure out how to merge projective geometry with the Cartesian coordinate system familiar to algebra students as a means of graphing equations. The system he developed, which uses what are called homogeneous coordinates, play a pivotal role in elliptic curve cryptography. Several decades later, in 1901, another French mathematician, Henri Poincaré, realized that points with rational coordinates—that is, points with coordinates that can be represented as fractions on the graph of an elliptic curve—composed a group. What Poincaré realized is that if you defined an operation (typically called 'addition') that took two rational points on the graph of the curve and yielded a third, the result was alwaysanother rational point on the curve. This process onlyworked if you used the homogeneous coordinates discovered by Möbius that include a point at infinity, however. Importantly, elliptic curve groups turned out to be Abelian, which meant that the order in which those addition operations were performed didn't matter. This is where matters stood until the mid-1980s, when Victor S. Miller, then a researcher at IBM, and Neal Koblitz of the University of Washington independently realized that you could build a public-private key cryptographic system based off elliptic curve groups. Encryption Keys Public-private key encryption, which is how almost all traffic on the Internet is secured, relies on two encryption keys. The first key, a private one, is not shared with anyone; it is kept securely on the sender's device. The second key, the public one, is composed from the private key, and this key is sent 'in the clear,' meaning that anyone can intercept it and read it. Importantly, both keys are required to decrypt the message being sent. In elliptic curve cryptography, each party agrees on a certain curve, and then each performs a random number of addition operations that start from the same point on the same curve. Each party then sends a number corresponding to the point they've arrived at to the other. These are the public keys. The other party then performs the same addition operations they used the first time on the new number they received. Because elliptic curve groups are commutative, meaning that it doesn't matter in what order addition is carried out, both parties will arrive at a number corresponding to the same final point on the curve, and this is the number that will be used to encrypt and decrypt the data. Elliptic curve cryptography is a relative latecomer to the encryption game. The first suite of tools did not appear until 2004, far too late to become a standard for the Web but early enough to adopted by the inventors of Bitcoin, which launched in 2009. Its status as the de facto standard for cryptocurrencies made people more familiar with it and more comfortable implementing it, although it still lags behind RSA encryption, the standard method in use today, by a wide margin. Yet elliptic curve cryptography has distinct advantages over RSA cryptography: it provides stronger security per bit and is faster than RSA. An elliptic curve cryptographic key of just 256 bits is roughly as secure as a 3,072-bit RSA key and considerably more secure than the 2,048-bit keys that are commonly used. These shorter keys allow for faster page rendering for Web traffic, and there's less processor load on the server side. Principles from elliptic curve cryptography are being used to try to develop cryptographic systems that are more quantum-resistant. If trends continue, the mathematics behind the vanishing point discovered by Renaissance artists 600 years ago may turn out to be a fundamental part of Internet encryption in the future.
