They are vast towers rising out of the jungles of the Amazon basin.

Tepuis, rocky tabletop mountains isolated by hundreds of metres of steep cliffs, form a unique habitat on their tops and inside their maze of caves. Untouched for millions of years, they host a secret world that has evolved in parallel with its surroundings.

Now, the secrets of the ancient tepui are slowly being revealed, thanks to expeditions led by speleologist of the University of Bologna, Italy.

Earlier this month, his team finished an arduous, 40-day expedition to some of the world鈥檚 last unexplored caves inside inaccessible tepuis. His team explored Imawar矛 Yeuta in Venezuela, which has at least 22 kilometres of tunnels, the largest known cave system of its kind.

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The caves are 鈥渁 completely different world鈥�, says Sauro. The quartz walls often have a spooky pinkish hue, and organic acids in the water turn the cave streams red.

The speleothems 鈥� stalactites and stalagmites 鈥� take fantastic forms. Some resemble billowing clouds of smoke; others look like a spray of mineral mushrooms. Exactly how they form is still a mystery.

Many of the peculiar, lumpy, silica speleothems that only occur inside quartzite caves are formed by colonies of microorganisms working together on the bacterial equivalent of a skyscraper.

Most known caves form in limestone 鈥� calcium carbonate 鈥� which readily dissolves in water.

But the tepui caves run through quartz sandstone, which is less susceptible to water erosion, so caves form much more slowly. Sauro notes that limestone caves form over hundreds of thousands or a few millions of years; his research suggests quartzite caves form over tens of millions of years.

鈥淲e don鈥檛 have a good idea of how old these caves are,鈥� says , a geographer and cartographer also at the University of Bologna. 鈥淭here鈥檚 nothing we can date inside, it鈥檚 all too old. It鈥檚 much older than what we expect.鈥�

Caves are like nature鈥檚 treasure chests, says de Waele. 鈥淭hey safeguard material from the outside 鈥� there鈥檚 no wind, no surface erosion,鈥� he says. The tepui caves have been protecting their contents for millions of years. 鈥淚t鈥檚 incredible.鈥�

Sauro says travelling the tepuis鈥� twists and turns is like taking a trip back in time. Each of his expeditions yields new discoveries. In 2012 the team turned up a never-before-identified . The team has also discovered new species of blind cave fish and bacteria.

Sauro is now focused on better understanding how life evolves in these sunless labyrinths, cut off from the rest of the world. The only way in is via entrances high in the walls or on the mountaintops, which can soar as high as 3000 metres.

The tepuis are isolated by their soaring cliff walls and home to unique species of animals. For example, Mount Roraima tepui is the only known habitat of the Roraima bush toad, a small amphibian that, in the face of danger, curls itself into a ball and rolls away.

The most recent expedition was Sauro鈥檚 sixth. The team鈥檚 discoveries won鈥檛 be released until November, but it鈥檚 likely they鈥檒l be able to introduce the world to another crop of previously unknown species.

Microbes that live in those caves are of special interest, says , a clinical microbiologist from the University of Queensland in Brisbane, Australia, who joined two of the expeditions. 鈥淐aves harbour chemical compounds and microbes that we don鈥檛 know about,鈥� he says.

By comparing these 鈥減ristine鈥� bacteria to modern superbugs, Zowawi aims to better understand how pathogens develop resistance to antibiotics. 鈥淲e hope that will widen our gaze about how those bacteria have evolved,鈥� he says.

His preliminary analyses have already turned up previously unclassified types of bacteria 鈥� but it鈥檚 too soon to know if they鈥檒l be useful.

There鈥檚 so much left to discover, says Sauro. Not only are quartzite caves and what鈥檚 hiding inside poorly understood, but individual tepuis can be very different. Each of the hundreds of mountains that dot the landscape developed and evolved independently of the others. 鈥淭hey are like islands in time,鈥� he says.

But it won鈥檛 be easy exploring them. You need a helicopter to get to the top, which depends on it being a clear day with minimal wind 鈥� in an area that is typically very cloudy and windy. Expeditions require a lot of money and waiting for near-perfect conditions.

Even once they鈥檝e disembarked, the explorers often encounter additional challenges. 鈥淭hey have cracks and canyons, and really rough terrain,鈥� says Sauro.

He is guided by satellite images that detect cave entrances on tepui tops 鈥� but those often end up being blocked by rocks and require a nervy descent on ropes. 鈥淥n most of them, there鈥檚 no access without climbing,鈥� says Sauro.

But he is undeterred and is already planning the next expedition for later this year.

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Read more: Hunt is on for world鈥檚 deepest caves more than 2km underground

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HOW low can you go? Dedicated deep cavers plumb the depths for an answer, and a newly announced expedition may just get to the bottom of it all.

For decades, cavers have competed informally to find the world鈥檚 deepest cave, pushing the boundaries of science along the way. It鈥檚 a perpetual quest 鈥� they can never truly know if they have found the absolute deepest. Bragging rights have bounced from cave to cave, but since 2004 the record has been , held by the Krubera Cave in the western Caucasus mountains, which is also home to the world鈥檚 deepest land animal, a springtail.

But Bill Stone may be about to break the record. The veteran caver and inventor has announced that he is planning a 2017 expedition to the Chev茅 Cave system, a sprawling supercave in the Oaxaca region of Mexico, whose tortuous tunnels underlie an area twice the size of Manhattan.

It鈥檚 not just the thrill of breaking records that drives exploration 鈥� scientific inquiry also matters. Caves are little-understood but important component of karst aquifers, which provide drinking water to hundreds of millions of people. Their steady climates also make them promising hunting grounds for well-preserved fossils, and mineral deposits can provide records of past changes in climate. For biologists, they promise to reveal new 鈥� and often bizarre 鈥� species.

鈥淭o me, and to most others on the team, Chev茅 is the world鈥檚 most fascinating exploration puzzle,鈥� says Stone. 鈥淚t鈥檚 the last terrestrial frontier.鈥�

Stone isn鈥檛 alone in thinking Chev茅 may break the depth record. 鈥淚t鈥檚 still possible that the deepest cave will be in the Americas, even deeper than Krubera,鈥� says Bill Steele, another veteran caver. He is now co-leading a series of expeditions to Sistema Huautla, a vast cave system that lies within sight of Chev茅, on the opposite side of a canyon carved by the Santo Domingo River. Sistema Huautla is the current deepest cave in North America and drops some 1545 metres from its entrances.

鈥淚t鈥檚 still possible that the deepest cave will be in the Americas, even deeper than the Krubera鈥�

Steele says Chev茅 has the 鈥減roven potential鈥� to capture the title. In 1990, caver James H. Smith, Jr, of dye into the water at one of its entrances. The green water emerged in the canyon and spilled into the Santo Domingo after dropping over 2547 metres (see also diagram). If Stone and his expedition can follow the water, Chev茅 will beat Krubera.FIG-mg30515101.jpg

But that鈥檚 a big challenge: while the dye suggests a path exists, it has to be traversed by people to beat the record. 鈥淚t doesn鈥檛 count if you just send a dye through a cave system鈥� any more than you can say Mount Everest was climbed the first time somebody saw the summit,鈥� says Steele.

Going deep into Chev茅 is a complex operation for which Stone says he will need between 60 and 70 cavers. His strategy is to look for a route past a boulder collapse that blocked further exploration back in 2003.

The appeal of the unknown and the element of surprise is what keeps Stone and others going. 鈥淵ou can still climb unclimbed peaks in the world, but you can also study them on Google Earth before you go,鈥� says cave diver Zeb Lilly. 鈥淵ou can dive the ocean depths, but those have been mostly mapped by sonar [and] it鈥檚 cost-prohibitive to do deep ocean exploration. On a budget, though, you can explore areas underground no one has ever seen.鈥�

And it鈥檚 not just depth that matters. Steele and fellow caver Tommy Shifflett have organised a series of annual expeditions to explore horizontal offshoots of the Sistema Huautla that past cavers ignored as they raced to the bottom. This year鈥檚 expedition was the second of 10 they have planned.

The expeditions have already turned up new species of tarantulas and scorpions, as well as bones thought to belong to extinct Pleistocene-era mammals, including a bison and a giant ground sloth. 鈥淚t鈥檚 a matter of being thorough, of doing all the various studies that should be done when you鈥檙e exploring an area never reached before by humans,鈥� says Steele.

(Image: Stephen Eginoire)

This article appeared in print under the headline 鈥淭he hunt is on for the deepest cave鈥�

(Image: A First rate Man-of-War driven onto a reef of rocks, floundering in a gale, Reinagle, George Philip/Royal Albert Memorial Museum/Bridgeman Images)

In 2007,听听at the US National Oceanic and Atmospheric Administration compiled a catalogue of听. Here are some of the most significant

1498听Columbus recounts how, on his third expedition to the Americas, a giant wave lifts up his boats during the night as they pass through a strait near Trinidad. Supposedly using Columbus鈥檚 words, to this day this area of sea is called the听Bocas del Drag贸n听鈥� the Mouths of the Dragon.

1853听The Annie Jane, a ship carrying 500 emigrants from England to Canada, is hit. Only about 100 make it to shore alive, to Vatersay, an island in Scotland鈥檚 Outer Hebrides.

1884听A rogue wave off West Africa sinks the Mignonette, a yacht sailing from England to Australia. The crew of four escape in a dinghy. After 19 days adrift, the captain kills the teenage cabin boy to provide food for the other three survivors.

1909听The steamship SS Waratah disappears without trace with over 200 people on board off the coast of South Africa 鈥� a swathe of sea now known for its high incidence of rogue waves.

1943听Two monster waves in quick succession pummel the Queen Elizabeth cruise liner as it crosses the North Atlantic, breaking windows 28 metres above the waterline.

1978听The German merchant navy supertanker MS M眉nchen disappears in the stormy North Atlantic en route from Bremerhaven to Savannah, Georgia, leaving only a scattering of life rafts and emergency buoys.

2001听Just days apart, two cruise ships 鈥� the Bremen and the Caledonian Star 鈥� have their bridge windows smashed by waves estimated to be 30 metres tall in the South Atlantic.

Read more about rogue waves, 9 modern ships they have smashed since 2006, and how we鈥檙e fighting to predict them: 鈥�Rogue waves: The real monsters of the deep鈥�

AT SOME point in their career every mathematician comes up against the question, is mathematics invented or discovered? The query makes some cranky. The answer doesn鈥檛 directly affect their work, after all, and the discussion often leads nowhere useful. Spending time debating the ultimate nature of mathematics takes away from actually doing it.

Some scholars take issue with the terms themselves. In his 2008 essay , Harvard University mathematician Barry Mazur called discovery and invention 鈥渢hose two too-brittle words鈥�. One might be tempted to defuse the question altogether with a merger: perhaps maths involves inventing new relationships between things we have discovered.

It鈥檚 a metaphysical query, a nerdy way to ask whether or not some pre-existing truths underlie our existence. Here we bump up against theology. If mathematical ideas are discovered 鈥� the Platonist position 鈥� then a proof is a real-world encounter with an immortal truth. But then where, exactly, is this ethereal pool of truths? Did prime numbers exist before the big bang?

If, on the other hand, mathematics is invented, then proofs spring from human intelligence a bit like art or law. But then why do mathematicians across time and space always agree on what鈥檚 right and wrong?

The question about invention versus discovery flares up every few years, often in a different guise. The latest incarnation concerns something very down to earth: money. In this case, the discovery versus invention question has profound consequences. In fact, there may be no mathematical question with higher stakes. That鈥檚 because mathematics powers the algorithms that drive computer software, and software is big business, worth over $300 billion a year to the global economy.

One of the most hotly contested issues in software is whether it should be patentable. At present not all of it can be, because of its dependence on mathematics. Patents are granted for inventions, and most mathematical formulas are deemed to be abstract ideas, not inventions, although the rules vary from country to country; some grant software patents, some don鈥檛. Where patents are granted, they are tantamount to big money.

鈥淢athematical formulas cannot be patented because they are deemed to be abstract ideas鈥�

Writing in the , David Edwards from the University of Georgia in Athens points out that the debate over patents boils down to the difference between discovery and invention. He argues that because abstract mathematical ideas are regarded as 鈥渄iscoveries鈥�, the system is fundamentally Platonic 鈥� and broken.

鈥淭here is no economic basis for the distinction between discovery and invention,鈥� he told me. 鈥淓conomically speaking, the difference between discovery and invention should be done away with.鈥� He argues patents should be granted for every new formula and algorithm, including those that power computer software. Such a change would jump-start innovation. 鈥淚f math were patentable, then you could have independent groups of mathematicians form a group or small company and support themselves that way,鈥� he says.

His position is extreme, but proponents of software patentability similarly argue that the system fuels growth and rewards people for their work.

Critics, notably the nonprofit (EFF) in San Francisco, say the current patent system actually stymies innovation, because getting a patent requires years and tens of thousands of dollars.

Another criticism is that hazy wording in software patent laws has given rise to 鈥減atent trolls鈥�, predatory companies that purchase old, cheap patents simply to use them to sue other companies for infringement. Trolling cost US companies $29 billion in 2011, according to a from the Boston University School of Law.

The Chicago-based company Soverain Software, for example, has been . It has racked up tens of millions of dollars suing companies including Amazon and Gap over patents for basic online shopping software. The online computer retailer Newegg, sued by Soverain for $2.5 million, fought back 鈥� and in January a US District Court in East Texas .

Even if trolls get banished, weak patents exist. The EFF argues that the US Patent Office fails to identify true inventions, resulting in 鈥渁 flood of bad patents on so-called inventions that are unoriginal, vague, overbroad, and/or so unclear that bad actors can easily use them to threaten all kinds of innovators鈥�. It has found patents issued for such basic tasks as sorting Facebook friends or purchasing a book with a single click 鈥� which are neither new nor original inventions. The beneficiaries of the system, they argue, are patent lawyers, not the general public.

Trolls and weak patents highlight the flaws in the current system 鈥� and they are keeping attention on the issue. Last month two US politicians introduced legislation that would penalise patent trolls. In the same month the US Patent and Trademark Office hosted two roundtable events 鈥� one at Stanford University in California, the other at New York University 鈥� to start discussions in the software world about how to improve the quality of patents.

The rallying cry of a good many critics remains 鈥渟oftware is mathematics鈥�, meaning that software shouldn鈥檛 be patentable. The odds are stacked against them, though 鈥� there鈥檚 too much money at stake. But if they lose, they can always console themselves that they may have inadvertently helped solve mathematics鈥� existential crisis.

As the world waits for the next installment of news in the hunt for the Higgs boson, a simple smart phone game encourages players to hide, not hunt, the world鈥檚 favourite particle.

Last December, researchers at the Large Hadron Collider at CERN near Geneva, Switzerland, reported that they had seen a hints of the Higgs boson, an elementary but still hypothetical particle that may help explain how mass forms.

Since then, more data has been collected, which could either strengthen or refute the tentative signal. These are due to be presented on 4 July at a , ahead of a hotly anticipated conference in Melbourne, Australia.

The game, , turns the quest on its head. Though physicists still hunt the Higgs, they are the bad guys 鈥� the player鈥檚 goal is to keep the particle hidden.

Hide that Higgs

Created by of Cambridge, Massachusetts, Agent Higgs depicts the slippery particle as a hapless and immobile secret agent wearing a ridiculous moustache. Other elementary particles also feature, including electrons, muons, other bosons and neutrinos.

Agent Higgs has a simple interface. Players shuffle these particles around a grid, trying to land on 鈥� and therefore conceal 鈥� the stationary Higgs.

Whether or not you can move a particle to a particular spot, however, is governed by the standard model of physics: our leading explanation of fundamental particles and force. That means electrons can鈥檛 approach each other because the electromagnetic force pushes them apart. And when a particle meets its antimatter doppelganger, both of them vanish. As the levels increase, it becomes more difficult to orchestrate the particles and their forces to keep the Higgs hidden.

Be warned: 鈥淎gent Higgs鈥� may be addictive. New 女生小视频 sent a link to the game to several physicists, and at least one is already hooked. 鈥淚 just wasted 2 hours on a sunny Saturday morning on it,鈥� says of Indiana University in Bloomington, who works on ATLAS, one of the main particle detectors at CERN.

Strong force

She has one gripe with the science however: 鈥淎n antineutrino meeting an electron should form a W boson 鈥� but who cares? It is a fun game and all the different rules certainly add to the fun.鈥�

A former educator at the before going full time as a game designer, Hall has previously designed games about gravity and special relativity. He鈥檚 currently expanding Agent Higgs to include quarks and the strong force, which holds quarks together in protons and neutrons, and holds protons and neutrons together in the nuclei of atoms.

鈥淭he strong force is not only the toughest one in particle physics to understand, but it will also make for the most difficult game play,鈥� he says.

There鈥檚 another major change he may have to consider in the near future: when physicists find the elusive particle 鈥� or rule out its existence 鈥� will Agent Higgs finally be out, making the game irrelevant? 鈥淚 don鈥檛 know,鈥� Hall says. 鈥淚鈥檓 going to have to track the progression of current events to see what happens. I鈥檒l have to wait and see.鈥�

The game costs and is designed for smart phones and tablets, although a stripped down version can be played on a computer.

Everyone wants a piece of pi (Image: Kimmo Taskinen/Rex Features)

The mathematical constant pi continues to infinity, but an extraordinary lawsuit that centred on this most beloved string of digits has come to an end. Appropriately, the decision was made on Pi Day.

On 14 March, which commemorates the constant that begins 3.14, US district court judge Michael H. Simon dismissed a claim of copyright infringement brought by one mathematical musician against another, who had also created music based on the digits of pi.

鈥淧i is a non-copyrightable fact, and the transcription of pi to music is a non-copyrightable idea,鈥� Simon wrote in his legal opinion dismissing the case. 鈥淭he resulting pattern of notes is an expression that merges with the non-copyrightable idea of putting pi to music.鈥�

The bizarre tale began about a year ago, when Michael Blake of Portland, Oregon, released a song and YouTube video featuring an original musical composition, 鈥淲hat pi sounds like鈥�, translating the constant鈥檚 first few dozen digits into musical notes. On Pi Day 2011, the number of page views skyrocketed as the video went viral, New 女生小视频 was among those who covered his creation. Blake found himself a nerd celebrity, fielding emails and phone calls from multiple media outlets.

Pi symphony

鈥淚t was a great morning,鈥� Blake recalls. 鈥淚t was the first time where something I鈥檇 done creatively received attention like that.鈥�

The celebratory ride quickly derailed, though. That afternoon, jazz musician from Omaha, Nebraska, cried foul. Erickson thought Blake鈥檚 work sounded suspiciously similar to his own 1992 piece 鈥淧i Symphony,鈥� also based on the digits of pi, which is registered with the US copyright office. He contacted YouTube, and Blake鈥檚 video vanished.

鈥淚t was like being stabbed,鈥� says Blake. 鈥淭his great thing I鈥檇 created, and then watched explode, was gone. I felt robbed.鈥�

Erickson and Blake, who have never met nor even talked on the phone, had both assigned each of the digits 0 to 9 to a musical note and then treated the digits of pi as a musical score. Erickson, who calls the two melodies 鈥渋dentical鈥�, filed a lawsuit claiming copyright infringement.

Judge Simon, however, disagreed. According to his ruling, the two pieces differed enough in areas like tempo, musical phrasing, and harmonies to be considered distinct. Plus, US law doesn鈥檛 protect every aspect of the piece, like underlying facts and ideas.

Copyrighted digits?

What鈥檚 more, Simon, who intentionally released his decision on Pi Day, noted that Erickson鈥檚 copyright registration only protects musical flourishes 鈥� and his are markedly different from Blake鈥檚.

Erickson isn鈥檛 happy though. 鈥淚f people look at my explanation of the Pi Symphony video and then look at Mr Blake鈥檚 video, they can draw their own conclusions,鈥� Erickson says. 鈥淚鈥檓 not sure that the judge got it right.鈥�

Blake says he wasn鈥檛 surprised by the ruling, but he still felt relief. For the last year, he鈥檚 been consulting multiple lawyers 鈥� working pro bono 鈥� in an effort to defend himself. 鈥淚t was great news to get on Pi Day,鈥� he says.

Stephen Joncus from the law firm of Klarquist Sparkman, LLP, also based in Portland, Oregon, doesn鈥檛 think there are any broader implications of the case in terms of copyright. 鈥淚 think it鈥檚 pretty standard for cases like this to be dismissed,鈥� he says. 鈥淵ou can鈥檛 get a copyright for an idea, and the idea here was making music based on the sequences of digits in the number pi.鈥�

Golden music

Both musicians continue to create mathematics-inspired musical pieces. Erickson鈥檚 latest creation explores powers of two, while Blake penned a piece last year that turns the digits of tau 鈥� a constant that at twice pi (6.28318) could replace pi 鈥� into music. He released that video on tau day, 28 June.

Blake plans to release his latest composition, a melodious phi (0.618鈥�) or , so-called because architects and artists find these proportions pleasing, on, you鈥檝e guessed it, 18 June.