Curves Of Constant Width

Curves of constant width

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The width of a circle is constant: its diameter.

But the circle is not the only shape that holds this pristine title. For instance let’s look at the Reuleaux triangle

Reuleaux triangle

A Reuleaux triangle is a shape formed from the intersection of three circular disks, each having its center on the boundary of the other two.

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The Reuleaux triangle is the first of a sequence of Reuleaux polygons, curves of constant width formed from regular polygons with an odd number of sides.

Some of these curves have been used as the shapes of coins

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To drill square holes.

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They are not entirely square, their edges are fillets i.e the edges are rounded and not sharp.

This animation offers a good insight as to why that is so.

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And in china, apparently on bicycles.

The man Guan Baihua shows his self-made multi-angle-wheel bicycle on May 6, 2009 in Qingdao of Shandong Province, China. Guan Baihua spent 18 months to complete this strange bicycle.

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Other shapes of constant width

There are other shapes of constant width beside the Reuleaux triangle ( that has been discussed in this post ), a whole bunch of them really. Do take a look at them. ( links below )

I will leave you guys with my favorite one.

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More:

If this post fascinated you, i strongly suggest you check these out. They go in-depth with the mathematics that underlies these curves and talk about other cool stuff:

An animation of non-circular rollers

Shapes and Solids of Constant Width - Numberphile  

Shapes of constant width

Reuleaux Polygons,           

Edit:

For those who are wondering if these are something that one would stumble upon on a regular basis. You may not find perfect ones but similiar ones definitely.

I found mine on a really old BMI calculator thingy. ( not sure what you would call it )

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Have fun exploring !

More Posts from Philosophical-amoeba and Others

8 years ago

Artist profile: Gerard Sekoto

‘Art is the spark, the illumination which is socially significant for it brings about understanding’ – Gerard Sekoto (1913–1993)

Artist Profile: Gerard Sekoto

Gerard Sekoto was born in Botshabelo, Mpumalanga province, in 1913, the year in which the Natives Land Act dispossessed many black South Africans of their ancestral lands. In 1938 Sekoto moved to Sophiatown, Johannesburg. He held his first solo exhibition the following year, and in 1940 the Johannesburg Art Gallery purchased his work Yellow Houses – A Street in Sophiatown (1939–1940). It was the first painting by a black South African artist to be acquired by a South African art institution, although Sekoto had to pose as a cleaner to see his own painting hanging in the gallery.

Artist Profile: Gerard Sekoto

Sekoto based this painting, titled Song of the Pick (1946), on a photograph taken in the 1930s of black South African workers labouring under the watchful eye of a white foreman standing behind them. However, in his painting the dynamic has changed. Sekoto has enhanced the grace and power of the labourers, turning them to confront the small and puny figure of the overseer, who appears about to be impaled by their pickaxes.

Sekoto painted this work in the township of Eastwood in Pretoria, shortly before moving to Paris in what became a lifelong exile from South Africa. During the 1980s, postcard-sized reproductions of this iconic painting were widely distributed in South Africa, as both a badge of honour and a source of inspiration in the struggle against apartheid.

Explore a diverse range of art stretching back 100,000 years in our exhibition South Africa: the art of a nation (27 October 2016 – 26 February 2017).

Exhibition sponsored by Betsy and Jack Ryan

Logistics partner IAG Cargo

Song of the Pick, 1946. Image © Iziko Museums of South Africa, Art Collections, Cape Town. Photo by Carina Beyer.

Song of the Pick was based on this image, taken by photographer Andrew Goldie in the 1930s.


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9 years ago
“Brainprint” Biometric ID Hits 100% Accuracy

“Brainprint” Biometric ID Hits 100% Accuracy

Psychologists and engineers at Binghamton University in New York say they’ve hit a milestone in the quest to use the unassailable inner workings of the mind as a form of biometric identification. They came up with an electroencephalograph system that proved 100 percent accurate at identifying individuals by the way their brains responded to a series of images. But EEG as a practical means of authentication is still far off.

Many earlier attempts had come close to 100 percent accuracy but couldn’t completely close the gap. “It’s a big deal going from 97 to 100 percent because we imagine the applications for this technology being for high-security situations,” says Sarah Laszlo, the assistant professor of psychology at Binghamton who led the research with electrical engineering professor Zhanpeng Jin.

Perhaps as important as perfect accuracy is that this new form of ID can do something fingerprints and retinal scans have a hard time achieving: It can be “canceled.”

Fingerprint authentication can be reset if the associated data is stolen, because that data can be stored as a mathematically transformed version of itself, points out Clarkson University biometrics expert Stephanie Schuckers. However, that trick doesn’t work if it’s the fingerprint (or the finger) itself that’s stolen. And the theft part, at least, is easier than ever. In 2014 hackers claimed to have cloned German defense minister Ursula von der Leyen’s fingerprints just by taking a high-­definition photo of her hands at a public event.

Several early attempts at EEG-based identification sought the equivalent of a fingerprint in the electrical activity of a brain at rest. But this new brain biometric, which its inventors call CEREBRE, dodges the cancelability problem because it’s based on the brain’s responses to a sequence of particular types of images. To keep that ID from being permanently hijacked, those images can be changed or re-sorted to essentially make a new biometric passkey, should the original one somehow be hacked.

CEREBRE, which Laszlo, Jin, and colleagues described in IEEE Transactions in Information Forensics and Security, involves presenting a person wearing an EEG system with images that fall into several cate­gories: foods people feel strongly about, celebrities who also evoke emotions, simple sine waves of different frequencies, and uncommon words. The words and images are usually black and white, but occasionally one is presented in color because that produces its own kind of response.

Each image causes a recognizable change in voltage at the scalp called an event-related potential, or ERP. The different categories of images involve somewhat different combinations of parts of your brain, and they were already known to produce slight differences in the shapes of ERPs in different people. Laszlo’s hypothesis was that using all of them—several more than any other system—would create enough different ERPs to accurately distinguish one person from another.

The EEG responses were fed to software called a classifier. After testing several schemes, including a variety of neural networks and other machine-learning tricks, the engineers found that what actually worked best was a system based on simple cross correlation.

In the experiments, each of the 50 test subjects saw a sequence of 500 images, each flashed for 1 second. “We collected 500, knowing it was overkill,” Laszlo says. Once the researchers crunched the data they found that just 27 images would have been enough to hit the 100 percent mark.

The experiments were done with a high-quality research-grade EEG, which used 30 electrodes attached to the skull with conductive goop. However, the data showed that the system needs only three electrodes for 100 percent identification, and Laszlo says her group is working on simplifying the setup. They’re testing consumer EEG gear from Emotiv and NeuroSky, and they’ve even tried to replicate the work with electrodes embedded in a Google Glass, though the results weren’t spectacular, she says.

For EEG to really be taken seriously as a biometric ID, brain interfaces will need to be pretty commonplace, says Schuckers. That might yet happen. “As we go more and more into wearables as a standard part of our lives, [EEGs] might be more suitable,” she says.

But like any security system, even an EEG biometric will attract hackers. How can you hack something that depends on your thought patterns? One way, explains Laszlo, is to train a hacker’s brain to mimic the right responses. That would involve flashing light into a hacker’s eye at precise times while the person is observing the images. These flashes are known to alter the shape of the ERP.


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7 years ago
Fangtooth Moray Photos By Sacha Lobenstein
Fangtooth Moray Photos By Sacha Lobenstein
Fangtooth Moray Photos By Sacha Lobenstein

fangtooth moray photos by Sacha Lobenstein


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7 years ago
Back When I Was A Studying Biology, I Noticed That A Lot Of Anatomical Terms Sound Like They Come Straight

Back when I was a studying biology, I noticed that a lot of anatomical terms sound like they come straight from Middle Earth. So, to celebrate the release of the last Hobbit film, I’ve created this INCREDIBLY nerdy quiz.

Do these words and phrases refer to parts of the human body, or reference people and places from J. R. R. Tolkien’s work?

Antrum of Highmore

Crypt of Morgagni

Caves of Androth

Lobelia

Loop of Henle

Scapha

Great Vein of Galen

Halls of Mandos

Groin

Gap of Calenardhon

Macewen’s Triangle

Canal of Schlemm

Gerontius

Islets of Langerhans

Meckel’s Cave

Chamber of Mazarbul

You shall not pass.


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8 years ago

Why do men have Adam's apples. It's so sexy but seems so unnecessary

It basically is unnecessary lol

It’s not that only men have Adam’s apples, women do too, but it’s less prominent. An Adam’s apple is just a piece of cartilage that protects your larynx (the voicebox) directly behind it. As boys & girls go through puberty, our voicebox grows which:

1. Causes our voice to deepen 2. Pushes the cartilage further forwards

In boys, the larynx grows in size significantly more which therefore pushes forward the evident bump of cartilage we like to call an Adam’s Apple.


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8 years ago

The Normans - A Timeline

911: According to later writer Dudo of Saint-Quentin, in this year the king of the Franks, Charles the Simple, grants land around the city of Rouen to Rollo, or Rolf, leader of the Vikings who have settled the region: the duchy of Normandy is founded. In return Rollo undertakes to protect the area and to receive baptism, taking the Christian name Robert.

1002: Emma, sister of Duke Richard II of Normandy, marries Æthelred (‘the Unready’), king of England. Their son, the future Edward the Confessor, flees to Normandy 14 years later when England is conquered by King Cnut, and remains there for the next quarter of a century. This dynastic link is later used as one of the justifications for the Norman conquest.

1016: A group of Norman pilgrims en route to Jerusalem are ‘invited’ to help liberate southern Italy from Byzantine (Greek) control. Norman knights have already been operating as mercenaries here since the turn of the first millennium, selling their military services to rival Lombard, Greek and Muslim rulers.

1035: Having ruled Normandy for eight years, Duke Robert I falls ill on his return from

a pilgrimage to Jerusalem and dies at Nicaea. By prior agreement, Robert is succeeded by his illegitimate son William, the future Conqueror of England, then aged just seven or eight. A decade of violence follows as Norman nobles fight each other for control of the young duke and his duchy.

1051: Duke William visits England. His rule in Normandy now established, and newly married to Matilda of Flanders, William crosses the Channel to speak with his second cousin, King Edward the Confessor of England. The subject of their conference is unknown, but later chroniclers assert that at this time Edward promises William the English succession.

1059: Pope Nicholas II invests the Norman Robert Guiscard with the dukedoms of Apulia, Calabria and Sicily. The popes had opposed the ambitions of the Normans in Italy, but defeat in battle at Civitate in southern Italy in 1053 had caused them to reconsider. In 1060 Robert and his brother Roger embark on the conquest of Sicily, and Roger subsequently rules the island as its great count.

1066: Edward the Confessor dies on 5 January, and the throne is immediately taken by his brother-in-law Harold Godwinson, the most powerful earl in England, with strong popular backing. Harold defeats his Norwegian namesake at Stamford Bridge in September. But on 14 October William’s Norman forces defeat Harold’s army at Hastings. William is crowned as England’s king on Christmas Day.

1069: The initial years of William’s reign in England are marked by almost constant English rebellion, matched by violent Norman repression. In autumn 1069 a fresh English revolt is triggered by a Danish invasion. William responds by laying waste to the country north of the Humber, destroying crops and cattle in a campaign that becomes known as the Harrying of the North, leading to widespread famine and death.

1086: Worried by the threat of Danish invasion, at Christmas 1085 William decides to survey his kingdom – partly to assess its wealth, and partly to settle arguments about landownership created by 20 years of conquest. The results, later redacted and compiled as Domesday Book, are probably brought to him in August 1086 at Old Sarum (near Salisbury), where all landowners swear an oath to him.

1087: William retaliates against a French invasion of Normandy. While attacking Mantes he is taken ill or injured – possibly damaging his intestines on the pommel of his saddle – and retires to Rouen, where he dies on 9 September. Taken to Caen for burial, his body proves too fat for its stone sarcophagus, and bursts when monks try to force it in. His eldest surviving son, Robert Curthose, becomes duke of Normandy, while England passes to his second son, William Rufus.

1096: Following a call to arms by Pope Urban II in 1095, many Normans set out towards the Holy Land on the First Crusade, determined to recover Jerusalem. Among them are Robert Curthose, who mortgages Normandy to his younger brother, William Rufus, and William the Conqueror’s notorious half-brother, Bishop Odo of Bayeux. Odo dies en route and is buried in Palermo, but Robert goes on to win victories in Palestine and is present when Jerusalem falls.

1100: Having succeeded his father in 1087 and defeated Robert Curthose’s attempts to unseat him, the rule of William II (‘Rufus’, depicted below) seems secure. But on 2 August 1100, while hunting in the New Forest with some of his barons, William is struck by a stray arrow and killed. His body is carted to Winchester for burial, and the English throne passes to his younger brother, Henry, who is crowned in Westminster Abbey just three days later.

1101: Roger I of Sicily dies. By the end of his long rule, Count Roger has gained control over the whole of Sicily – the central Muslim town of Enna submitted in 1087, and the last emirs in the southeast surrendered in 1091. He is briefly succeeded by his eldest son, Simon, but the new count dies in 1105 and is succeeded by his younger brother, Roger II.

1120: On 25 November Henry I sets out across the Channel from Normandy to England. One of the vessels in his fleet, the White Ship, strikes a rock soon after its departure, with the loss of all but one of its passengers. One of the drowned is the king’s only legitimate son, William Ætheling. Henry responds by fixing the succession on his daughter, Matilda, and marrying her to Geoffrey Plantagenet, count of Anjou.

1130: Roger II is crowned king of Sicily, having pushed for royal status in order to assert his authority over the barons of southern Italy. A disputed papal succession in 1130 has provided an opportunity and, in return for support against a papal rival, Pope Anacletus II confers the kingship on Roger in September. He is crowned in Palermo Cathedral on Christmas Day.

1135: Henry I dies in Normandy on 1 December, reportedly after ignoring doctor’s orders and eating his favourite dish - lampreys. His body is shipped back to England for burial at the abbey he founded in Reading. Many of his barons reject the rule of his daughter, Matilda, instead backing his nephew, Stephen, who is crowned as England’s new king on 22 December.

1154: King Stephen, the last Norman king of England, dies. His death ends the vicious civil war between him and his cousin Matilda that lasted for most of his reign. As a result of the Treaty of Wallingford, which Stephen was pressured to sign in 1153, he is succeeded by Matilda’s son Henry of Anjou, who takes the throne as Henry II.

1174: King William II of Sicily begins the construction of the great church at Monreale (‘Mount Royal’), nine miles from his capital at Palermo. The building is a fusion of Byzantine, Latin and Muslim architectural styles, and is decorated throughout with gold mosaics, including the earliest depiction of Thomas Becket, martyred in 1170.

1194: Norman rule on Sicily ends. Tancred of Lecce, son of Roger III, Duke of Apulia, seizes the throne on William’s death in 1189; on his death in 1194 he is succeeded by his young son, William III. Eight months later, Holy Roman Emperor Henry VI, husband of Roger II’s daughter Constance, invades Sicily and is crowned in Palermo on Christmas Day. The following day, Constance gives birth to their son, the future Frederick II.

1204: King John loses Normandy to the French. The youngest son of Henry II, John had succeeded to England, Normandy, Anjou and Aquitaine after the death of his elder brother, Richard the Lionheart, in 1199. But in just five years he lost almost all of his continental lands to his rival King Philip Augustus of France – the end of England’s link with Normandy.


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7 years ago

First genetic proof that women were Viking warriors

First Genetic Proof That Women Were Viking Warriors

New DNA evidence uncovered by researchers at Uppsala University and Stockholm University shows that there were in fact female Viking warriors. The remains of an iconic Swedish Viking Age grave now reveal that war was not an activity exclusive to males – women could be found in the higher ranks at the battlefield.

The study was conducted on one of the most well-known graves from the Viking Age, a mid-10th century grave in Swedish Viking town Birka. The burial was excavated in the 1880s, revealing remains of a warrior surrounded by weapons, including a sword, armour-piercing arrows, and two horses. There was also a full set of gaming pieces and a gaming board.

The morphology of some skeletal traits have long suggested that she was a woman, but since this grave has been the type specimen for a Viking warrior for over a century, it has always been assumed to have belonged to a male Viking. Now, geneticists, archaeogeneticists and archaeologists have worked together and solved the mystery. DNA retrieved from the skeleton demonstrates that the individual carried two X chromosomes and no Y chromosome. Read more.


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9 years ago

Mary Anning

Mary Anning

Mary Anning was born on 21 May in 1799, one of the most famous fossil finders of her day. This large skull was collected by Anning and is part of the skull and lower jaw of an ichthyosaur (Ichthyosaurus platyodon).

Mary Anning’s family had earned a living for years by gathering fossils on the shore at Lyme Regis in Dorset to sell to collectors. Mary learned about the fossils from her parents, Richard and Mary (‘Molly’) Anning. Despite the lack of a formal education, Mary Anning became an expert on the fossils she found, and the most eminent geologists of the day often sought her advice. In the 1820s she became the first person in Britain to find complete specimens of an ichthyosaur, a plesiosaur and a pterodactyl.

The specimens that Anning collected can still be found in museums throughout Britain. The British Museum purchased this example shortly after Anning discovered it.

On loan from the Natural History Museum.


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8 years ago
On This Day, 14th February 1779,  Captain James Cook Was Killed In Hawaii.
On This Day, 14th February 1779,  Captain James Cook Was Killed In Hawaii.

On this day, 14th February 1779,  Captain James Cook was killed in Hawaii.

James Cook completed three major voyages of discovery. On his first, departing in 1768, he commanded the ‘Endeavour’ on an expedition to chart the transit of Venus. He returned to England in 1771, having also circumnavigated the globe, including exploring and charting New Zealand and Australia’s eastern coast. 

On his second journey (1772-1775), he commanded the 'Resolution’ and the 'Adventure’ on an expedition to the South Pacific, disproving the rumour of a great southern continent, exploring the Antarctic Ocean, New Hebrides and New Caledonia.

Cook’s third and final voyage (1776-1779) of discovery was an attempt to locate a North-West Passage, an ice-free sea route which linked the Atlantic to the Pacific Ocean. Again, Cook commanded the Resolution while Charles Clerke commanded Discovery. Leaving England in 1776, Cook first sailed south to Tahiti to return Omai, a Tahitian man, to his home. Omai had been taken on Cook’s second voyage and had been an object of curiosity in London. It was on this, Cook’s final voyage, that he discovered the Hawaiian Islands in January 1778. This major discovery would lead to his death – Cook was killed on a return visit to Hawaii at Kealakekua Bay, on 14 February 1779.  

Kealakekua Bay was considered the sacred harbour of Lono, the fertility god of the Hawaiians. Cook and his compatriots were welcomed as gods but after one of the crewmen died, exposing the Europeans as mere mortals, relations became strained. On February 4, 1779, the British ships sailed from Kealakekua Bay, but rough seas damaged the foremast of the Resolution, and after only a week at sea the expedition was forced to return to Hawaii.

The Hawaiians greeted Cook and his men by hurling rocks; they then stole a small cutter vessel from the Discovery. Negotiations with King Kalaniopuu for the return of the cutter collapsed after a lesser Hawaiian chief was shot to death and a mob of Hawaiians descended on Cook’s party. The captain and his men fired on the angry Hawaiians, but they were soon overwhelmed, and only a few managed to escape to the safety of the Resolution. Captain Cook himself was killed by the mob. A few days later, the Englishmen retaliated by firing their cannons and muskets at the shore, killing some 30 Hawaiians. The Resolution and Discovery eventually returned to England.

The State Library of New South Wales holds significant original sources relating to James Cook, these paintings from the collection depict the death of Captain Cook.

On This Day, 14th February 1779,  Captain James Cook Was Killed In Hawaii.

Carved ditty box shaped like a coffin on silver stand, containing a rough watercolour sketch of the death of Cook, including a lock of Cook’s hair, ca. 1779 / carved by sailors on Cook’s last ship HMS Resolution. State Library of NSW.


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9 years ago
Photographer Florent Tanet’s Precarious (non-photoshopped) ‘equilibrium’ Series
Photographer Florent Tanet’s Precarious (non-photoshopped) ‘equilibrium’ Series
Photographer Florent Tanet’s Precarious (non-photoshopped) ‘equilibrium’ Series

Photographer Florent Tanet’s precarious (non-photoshopped) ‘equilibrium’ series


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philosophical-amoeba - Lost in Space...
Lost in Space...

A reblog of nerdy and quirky stuff that pique my interest.

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