Tag Archives: symmetry


Mathematical concepts can be very useful for us to generate beauty. By using the trigonometric functions sine and cosine, we can make an infinite number of stunning symmetrical images. Below you can see seven images and the formulas I used to create them. Each of these shapes is constructed by 7,000 circles.

7,000 Circles (1)

Credit: Hamid Naderi Yeganeh

7,000 Circles (2)

Credit: Hamid Naderi Yeganeh

7,000 Circles (3)

Credit: Hamid Naderi Yeganeh

7,000 Circles (4)

Credit: Hamid Naderi Yeganeh

7,000 Circles (5)

Credit: Hamid Naderi Yeganeh

7,000 Circles (6)

Credit: Hamid Naderi Yeganeh

7,000 Circles (7)

Credit: Hamid Naderi Yeganeh

See more images at: https://mathematics.culturalspot.org

Extremely rare, unlikely crystal found in Russian meteorite

Scientists have discovered some of the most bizarre crystals in a piece of Russian meteorite. It’s only the third time such a crystal has been found in nature, all three samples coming from the same meteorite.

Image credit: Quasicrystal atomic structure (L), Talapin et al.

The crystals in case are called quasicrystals – structures that have the order characteristic to crystals but don’t share their periodicity. They’re remarkable for two reasons: firstly, they’re incredibly rare in nature and secondly, they’re incredibly unlikely. They’re so unlikely that they cost the scientist who first discovered them his job. Israeli chemist Daniel Shechtman, who first proposed their existence, was considered mad for a long time.

The reason for this seems fairly straightforward – crystals are typically filled with neat shapes like cubes or triangles (four-fold and three-fold symmetry respectively). Other shapes would leave a gap behind. But quasicrystals are arranged in irregular, five-fold symmetry, which should just not work. True story – a few years ago during my undergrad, we were taught that there’s no such thing as five-fold (pentagonal) symmetry.

Pentagonal symmetry. Image credits: J.W. Evans, Ames Laboratory, US Department of Energy

Patricia Thiel, a chemistry and materials science expert based at Iowa State University, used this helpful analogy of tiling a floor to explain the unique properties of quasicrystals in this NPR article:

“If you want to cover your bathroom floor, your tiles can be rectangles or triangles or squares or hexagons,” she said. “Any other simple shape won’t work, because it will leave a gap. In a quasicrystal, imagine atoms are at the points of the objects you’re using. What Danny [Shechtman] discovered is that pentagonal symmetry works.”

So now we know that the strict rules of symmetry can be broken, not only in the lab but also in nature. This sample was found by a team led by geologist Luca Bindi from the University of Florence in Italy.

“What is encouraging is that we have already found three different types of quasicrystals in the same meteorite, and this new one has a chemical composition that has never been seen for a quasicrystal,” one of the team, Paul Steinhardt from Princeton University, told Becky Ferreira at Motherboard.

“That suggests there is more to be found, perhaps more quasicrystals that we did not know were possible before.”

To make things even more interesting, this sample was new and undiscovered. The previous two had been synthesized in a lab, but the latest finding is brand new. It features icosahedral symmetry, an exotic pattern featuring 60 points of rotational symmetry, somewhat like that of a soccer ball.

But while quasicrystals are relatively simple to make in a lab nowadays, we still haven’t found that many in nature. Paul Steinhardt, who serves as the Albert Einstein professor at the science at Princeton University, was one of the authors of the new research. He says that one of the reasons we haven’t found any more of them is because almost no one is really looking.

“There are perhaps one or two other groups [of researchers] at most searching,” Steinhardt said. “Since we found an example, we have been focusing on understanding how this particular one formed since that will tell us something about the likelihood of finding others. But it is very very early times for these kinds of studies.”

The moon jellyfish move ther remaining limbs around to become symmetrical again. Image: Michael Abrams and Ty Basinge

Moon Jellyfish morphs back into symmetry after losing limbs

A novel, previously unseen self-repair mechanism was reported by a team of researchers at Caltech who studied the moon jellyfish. A lot of animals, mostly invertebrates, grow back their lost limbs after these are bitten off by predators or lost in an accident. The moon jellyfish, however, employs a different tactic altogether: instead of expending a lot of energy to regrow its lost limb, the animal re-arranges the limbs it has left to regain symmetry. Even when it’s left with two limbs out of its initial eight, the jellyfish will still re-arrange itself. This sort of mechanism might prove extremely useful in designing self-repairing robots.

Back in symmetry

The moon jellyfish. Image: Terra Spirit

The moon jellyfish. Image: Terra Spirit

Aurelia aurita or the moon jellyfish is one of the most common jellyfish species in the world. It’s translucent, usually about 25–40 cm and easily recognizable by its four horseshoe-shaped gonads visible at the center of the bell. It’s remarkable though that given the animal has been widely studied, it’s only recently that we learn of its unique self-repair ability.

“We’ve now observed another self-repair mechanism,” says researcher Michael Abrams of the California Institute of Technology (Caltech), a graduate student in biology and biological engineering. “It kind of broadens our definition, a little bit, of self-repair.”

Abrams and colleagues focused on larval moon jellies, known as ephyrae. During this stage, the juveniles only measure 1cm in diameter, but they’re limbs look and behave just as in the adult stage. Typically, the moon jellies are born with eight limbs arranged in a radial pattern.

Initially, the researchers wanted to see if the jellyfish could regrow its lost limbs, like other invertebrates. So they amputated one or several limbs from anesthetized ephyrae then introduced them to their familiar salt water environment. The jellyfish didn’t regrow the lost limbs, but instead behaved far more remarkably.

The moon jellyfish move ther remaining limbs around to become symmetrical again. Image: Michael Abrams and Ty Basinge

The moon jellyfish move ther remaining limbs around to become symmetrical again. Image: Michael Abrams and Ty Basinge

In the image above, the top row shows the process of symmetrization after losing four of eight limbs. The bottom row shows the same process after losing five of eight limbs. Basically, the symmetrization occurred with whatever limbs the jellyfish had left, even just with two.

“Pretty quickly, we realized that they were doing something very different than what anyone had ever talked about before,” Abrams says.

The jellyfish likely adapted this feature because symmetry is essential to its survival. The symmetrical limbs act like paddles which help the animal swim and pull food towards the central mouth. “And you can imagine how this paddling surface would be disturbed if you have a big gap between the arms,” Abrams said.

To study how the moon jellyfish re-arranges itself, the researchers used a cell proliferation stain to track cell death and birth. Even more surprising, the animals didn’t use cell growth or shrank parts to re-purpose itself. Instead, most likely the pulsing of the muscles is behind the mechanism. As the creatures swims about, it pulls the remaining arms into new positions. When they put this hypothesis to the test by applying muscle relaxers, the amputee jellyfish were unable to regain symmetry, as reported in PNAS.

Remember, all of this happen inside a creature with no brain! Robots, aren’t that far behind so this neat trick might be helpful. A damaged robot can’t grow back a mechanical part, put it can sure reconfigure itself to become functional again. The moon jellyfish might serve as an example.