curved surface


If you’ve ever bitten into a chocolate-covered bonbon, you may have noticed that the candy’s chocolate coating is remarkably uniform. Inspired by this observation, a group of engineers have investigated how viscous fluids poured over a curved surface flow and solidify; their findings were published this week. 

Rather than heated chocolate, the group used polymer-filled fluids that cure and harden over time. Interestingly, they found that the final shell is quite uniform and that its thickness does not depend on the pouring technique. Instead, they can predict the final shell thickness based on the radius of the mold and the rheological properties of the fluid–specifically its density, viscosity, and curing time. The reason for this is that the time it takes for the fluid to drain and coat the mold is much shorter than the time it takes for the polymer to cure. As a result, the amount of fluid that sticks to the mold depends on geometry and fluid properties - not how the fluid was poured. 

Amateur confectioners rejoice: pouring uniform chocolate coatings may be easier than you thought!  (Image credit: MIT News, video; research credit: A. Lee et al.)

From NSF Science360 Picture Of The Day; November 18, 2015:

Using Math To Predict Surface Patterns

A team of mathematicians and engineers at the Massachusetts Institute of Technology (MIT) has developed a mathematical equation that predicts how surface patterns form on curved objects. Pictured is a sphere with a combination of hexagons and labyrinthine patterns, and a more complex, torus-shaped object with hexagonal dimples. When a grape slowly dries and shrivels, its surface creases, ultimately taking on the wrinkled form of a raisin. Similar patterns can be found on the surfaces of other dried materials, as well as in human fingerprints. While these patterns have long been observed in nature, and more recently in experiments, scientists have not been able to come up with a way to predict how such patterns arise in curved systems, such as microlenses.

Visit Website | Image credit: Norbert Stoop

Boomerangs work in space. Because the path of a boomerang is the result of uneven forces on its curved surfaces as it flies through the air, the presence of gravity is irrelevant. As long as you’re in an oxygenated, enclosed environment, a boomerang will return to you even in zero gravity. Source


Delicately Handpainted Ceramics by Elizabeth Becker

Artist Elizabeth Becker from Teacupco runs a ceramic ware boutique where she features hand painted housewares which have elegant and colourful animals painted onto the porcelain. 

The animals are carefully emblazoned on the curved surface, using the three-dimensional space to the greatest advantage by smartly positioning the foxes, birds and rabbits frolicking, flying or curled into a ball at the base of the cup.  The painting using bold colours, defined and blunt, which is Teacupco’s trademark rustic artistry. The heartwarming images of the creatures are sure to provide a homely welcome to the warm brews stirring within and a fresh relief to the eye that the choice of colour fits into the counter tops of cosy cabins or contemporary kitchen decor because of its clever artistic imagery. Find them in their Etsy shop.

View similar posts here!


goats cannot overcome curved or slanted surfaces


A Sweeping Symbol of Modernity in Azerbaijan

For more photos and videos from The Heydar Aliyev Center, explore the Heydar Aliyev Center location page.

The Heydar Aliyev Center in Azerbaijan resembles a cresting wave—there are no straight lines on the structure’s curved, white surface. Constructed by British architect Zaha Hadid in 2012, the center’s unique shape is a symbol of modernity in the city of Baku, reflecting the present and the future in progress. Heydar Aliyev stands primarily as a venue for art exhibitions, and its stunning landscape is also popular backdrop for visiting and local Instagrammers alike.


San Francisco-based design professor and illustrator Miguel Cardona transforms ordinary paper coffee cups into bold works of art. Because of the curved surface of his canvas of choice, each piece is rendered freehand and he thoroughly enjoys the challenges that this presents:

“You have this three-dimensional object that is in your hands, you can pull the cup in a different direction and hold the pen still. You can also hide a lot of flawed perspective. You don’t need a desk, it can be done anywhere, and to protect it, you can stack it in another blank cup. The cup itself can hold your art supplies and is itself, a display stand, it’s quite the perfect design.”

Cardona’s subjects vary from pop culture character and icons to robots, monsters, and even bodily organs. But these beautiful illustrations aren’t quite as awesome as what he does with them. Miguel sells each finished piece for $20 and donates 100% of the proceeds to Project Night Night, which donates baby blankets, children’s books, and toys to children in homeless shelters.

Visit Miguel Cardona’s website to check out more of his fantastic illustrated coffee cups.

[via Design Taxi and Cool Hunting]

Ram’s Head Dagger

India (likely Jaipur), Mughal, 18th or 19th century

Hilt: Gold, enameled and set with precious stones; kundan technique Blade: steel

Often tucked into a sash or horseman’s boot, daggers in Mughal India displayed the wealth and power of their owners. An intricately patterned ram’s head pommel adorns the hilt of this dagger, made in the kundan technique in which gems are set into malleable pure gold foil, allowing them to be arranged in any pattern or density over curved surfaces. In this dagger, pieces of quartz adorning the cross guard are surrounded by raised borders of gold which form the curved lines of a flower. The ram’s head is decorated with a floral scroll and is separated from the hilt grip by a quartz collar, also in the kundan method.

This dagger bears a striking resemblance to another dagger posted recently.


The image is not mine. It is a fantastic creation by bigblueboo that has caught some attention outside of the usual math tumblverse. You should definitely check out eir blog and if you like this post you should (also?) reblog the original. With that out of the way:

Modeling. Mathematical modeling is the art of translating real systems, often physical or economic, into the language of mathematics in an attempt to predict future behavior of the system. Doing this often requires making many simplifying assumptions which are “unwarranted” from a purely logical perspective, but make the problems tractable. Therefore mathematical modeling is a distinct (but related) skill from the modern conception of the practice of mathematics.

A “parametric equation” is difficult to define exactly, but it is often (as it is here) a method for producing general surfaces or curves that cannot be described by functions because they do not pass the vertical line test. More specifically for this example, it is a function from a line segment into a higher-dimensional space.

This object has inspired me to get off my lazy butt and start producing content for the blog again. It also happens to be an excellent source of mathematical content: I’m probably going to be doing a daily series of posts about it for a while. I know I have content for at least three days and probably a few more besides.

It is vaguely related to epicycles, but the name “generalized epicycle” is my own invention. If you know an actually recognized name, I would love to know about it!

(EDIT: There is an old version of this post in which the constant terms were omitted and there were some flopped sin/cos symbols. I’m sorry! These haven’t been edited by someone else unlike many of the proofs I post so they’re bound to be a little rough around the edges every so often.)



The envelope of light rays reflected or refracted by a curved surface is called a Caustic.

All the light rays incident on the caustic surface are tangent to it.

The next time you take a sip from your coffee mug, be sure to take a look at its soul that it benevolently exhibits.

In the case of a circle, the caustic shape/curve formed is the cardioid.

Even water produces some really interesting caustic patterns

and so does wine in a glass.

Some of the images that have portrayed in the post have been made using computer graphic rendering systems. They trace the possible paths of light beam, accounting for refraction and reflection.

Have a great day!

Sources:   caustics  harvard  mr-cad


Jean François Niceron - The Artificial Magic in Optical Distortions, “La Perspective Curieuse ou Magie Artificielle des Effets Merveilleux de l'Optique” (The Curious Perspective or Artificial Magic from the Marvellous Effects of Optics), 1663.

Catoptrics is an area of study concerned with the properties of reflection and the formation of images by reflecting light off mirrors.
Dioptrics is a branch of optics dealing with the refraction of light, especially by lenses.
Anamorphosis is a distorted projection of an image which only becomes clear when the observer’s point of view changes or it is viewed as a reflection produced by a specific curved mirror surface.

These 17th century book illustrations instruct artists about the basic geometrical properties involved in producing artworks with some types of projected and distorted perspectives and optical illusions.


Desert-dwelling plants like cactuses have to be efficient collectors of water. Many types of cactus are particularly good at gathering water from fog that condenses on their spines. Droplets that form near a spine’s tip move slowly but inexorably toward the base of the spine so that the cactus can absorb them. The secret to this clever transport lies in the microstructure of the spine’s surface. The Gymnocalycium baldianum cactus, for example, has splayed scales along its spines. Capillary interactions with the scales result in differences in curvature on either side of the droplet. Curved fluid surfaces generate what’s known as Laplace pressure, with a tighter radius of curvature causing a larger Laplace pressure. Because the curvature of the droplet varies from the base side to the tip side of the spine, the difference in Laplace pressures across the droplet creates a force that drives the droplet toward the spine’s base. (Image credit: C. Liu et al., source)


Yves Saint Laurent Gloss Volupté: Voluptuous Kisses in 2014!

Seems every other high-end makeup junkie has a stash of the classic, melt-on-your-lips, gorgeously-packaged Rouge Volupté lipsticks in their drawer, so I’m expecting some of the fanaticism to continue in the new year with the launch of 24 brand new shades of Gloss Volupté. (Shade availability may vary by country, so check with your local counters for more information!)

If you like the plush, lightweight, balm-like feel of the lipsticks, you will probably also love the weightless “melting” feel of these new-generation glosses that give high shine in one light coat without needing to glop on a thick layer of product. No gummy, greasiness or stickiness! 

The adorable lip-shaped applicators come with a hollow well in the center, and a gentle bend, to pick up just the right amount of product, curve over the lip surface comfortably, and distribute product evenly.

I received 4 shades for reviewing/swatching purposes, so unfortunately I won’t be able to show you every single color but the names and numbers are all listed below for reference. 

Gold - 10 shades with yellow-gold micro-sparkles

N° 1 GOLD* (translucent gold sparks)










Iridescent - 5 shades with a metallic shine





N° 105 BRUN LAMÉ – Limited Edition

Pure - 9 shades with a translucent, shimmer/sparkle-free creme finish

N° 201 PURE* (clear)







N° 208 FAUVE

N° 209 SMOKING* – Limited Edition (translucent black)


These three shades are designed to be applied as a top coat over another GLOSS VOLUPTÉ or a ROUGE VOLUPTÉ lipstick for intense sheen or a special effect.

(Pictured at top: YSL Rouge Volupté Shine N° 16 Orange Impertinent, with Gloss Volupté N° 1 GOLD, N° 19 ROSE ORFÈVRE, N° 202 ROSE JERSEY and N° 206 FUCHSIA ORAN.)