Pi – Discovering the Mathematical Brilliance Behind Gauss’s Method and its Modern-Day Application” We’ve been there. You grab an iceberg of pizza, and are planning to eat it however, it falls over and hangs loosely off the fingers.
The crust isn’t strong enough to hold it’s weight. pizza. Perhaps you should have opted with fewer toppings. However, there’s no reason to be discouraged, because the years of pizza eating will have taught you how to handle this scenario.
Simply fold your pizza into the shape of a U form (aka fold hold) and then place it in a slant (aka fold hold). The fold holds the slice back from slipping across the table, and you can go on enjoying your food. (If the slice isn’t there pizza in your kitchen, test this using a paper.)
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The secret behind this pizza trick is an astonishing mathematical proof of curves, one that’s so shocking that its creator the mathematic brilliant Charles Friedrich Gauss has named this result the Theorema Egregium, Latin for extraordinary or astonishing theorem.
Make a paper sheet and make it into a cylinder. It may seem evident it’s flat but the cylinder is curving. Gauss thought about this in a different manner. He was looking determine the curve an object in a way that isn’t affected by the way your bend it.
If you focus your attention on an ant who lives on the circle, there are many possibilities for the ant to follow. It might choose to follow the curving path, drawing out a circle. Or it could follow the flat path and trace out an unidirectional line. Then it could take a different route and trace out the spiral.
Gauss’s genius idea was to define curvature of the surface in a way that takes all of these options into consideration. Here’s how. Beginning at any point, identify the two extreme paths that an ant could take (i.e. the most concave and the convex one).
Then, multiply the curvature curve of both paths (curvature can be positive in concave pathways and zero for flat paths while it is negative on convex ones). The number you will get is Gauss’s definition for the curvature in that area.
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Let’s look at some of the examples. For the ant in the cylinder, two paths that are available to it are the curving, circle-shaped path, as well as the straight, flat path. However, since the flat line has no curvature, if you multiply both curvatures by zero, you will receive zero.
Mathematicians say that a cylinder is flat since it has no Gaussian curvature. This is due to the fact that it is possible to make one from the paper.
If instead, the ant was the surface of a ball, there would have no flat paths to it. Each path is curving by equal amounts, consequently the Gaussian curvature is a positive number. Therefore, spheres are curving while cylindrical objects are flat. It is possible to bend a piece made of newspaper into a tube however, you cannot create the shape of a ball.
Gauss’s most remarkable theorem, one that I would like to imagine caused him to laugh in delight the organism living on a flat surface can determine its curvature, without needing to leave the surface, simply by observing distances and doing some math.
This, of course is the reason we are able to know if our universe is curving, without ever needing to venture outside of the universe (as as we can discern, it’s flat).
The most surprising consequence is you can take a flat surface and bend it in any way you’d like, as that you do not stretch, shrink, or tear it. Likewise, the Gaussian curvature remains unchanged.
It’s because bending does not alter the distances that are visible on the surface. Therefore, the ant that lives on the surface will determine using the exact Gaussian curvature that it did before.
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This may sound abstract, but it actually has practical implications. Slice an orange into two halves, and eat the flesh (yum) and then put this dome-shaped piece of fruit on your ground and then pound on it. The peel won’t become flat as it stretches out into the shape of a circle. Instead, it’ll break apart.
This is because a sphere and a flat surface have distinct Gaussian curvatures. There’s any way for a flat surface to be more evenly shaped with out distorting or breaking it.
Have you ever attempted wrapping gifts around for a basketball? The same issue. Whatever way you stretch a piece of paper, it will always retain the remnant of its flatness, and you’ll get a crinkled mess.
A further consequence of Gauss’s Theorem is that it’s not possible to accurately represent maps on paper. The world map you’ve come to know illustrates angles in a correct manner, but it distorts the areas. Museum of Math Museum of Math highlights that designers for clothing face the same problem designing designs on flat surfaces which must fit our curvaceous bodies.
What does this have to anything to do with pizza? It’s true that the pizza was flat before you took it home (in math terms that is, it has no Gaussian curvature). Gauss’s awe-inspiring theorem reassures that the direction of the pizza must remain flat regardless of the way you bend it the pizza must maintain an indication of its flatness from its initial state.
If the slice is flipped across, its flat side (shown as red in the below) is pointing sideways and isn’t ideal to eat it. However, by fold the slice upside down by bending it, you force it to be flat in the opposite direction – the one that is directed towards your mouth. Theorema egregium, indeed.
When you bend a sheet in one direction then you force it to stiffen in the opposite direction. Once you realize this idea and begin to see it all over the place. Take a close look at the grass blade.
It’s typically folded in the central vein. This increases its stiffness and keeps it from slipping over. Engineers commonly utilize curvature to increase the strength to their structures.
For instance, at the Zarzuela course located in Madrid in Spain, the Spanish structural engineer Eduardo Torroja developed an innovative concrete roof that extends from the stadium and covers the entire area and only two inches thick. It’s a pizza trick in disguise.
Curvature is a source of the strength. Imagine this: you could sit on a soda bottle and it’ll effortlessly support the weight of your body. But the walls of this container is only one thousandth inches thick.
That’s roughly the same thickness as a piece of paper. The secret behind the incredible stiffness of a soda bottle is the curvature. This can be demonstrated when someone pokes the soda container with pencil when the can is on your feet. Even if you only make a tiny scratch, the can will massively buckle to your weight.
The most obvious instance of strength by curvature is the ubiquitous construction materials made of corrugated (corrugate is derived from the word ruga which is Latin which means wrinkle).
It’s hard to imagine anything less boring than an corrugated, cardboard container. You can tear the box in half and you’ll see an unmistakable, wavy layer of cardboard within the walls.
The wrinkles aren’t to enhance the aesthetics. They’re a clever method to make a material thin and light, but strong enough to withstand bending under the weight of a great deal of force.
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Corrugated sheets of metal make use of the same concept. These basic, simple products are an expression of utility in their design perfectly in line with their purpose. Their strong and durable properties as well as their inexpensive cost have incorporated with the rest of the modern world.
Nowadays, we don’t think about these wrinkly sheets of metal another thought. However, when it first introduced, people saw corrugated iron as an amazing material. It was first patented around 1829, by Henry Palmer, an English engineer responsible for building the London Docks.
Palmer constructed the world’s first iron corrugated structure known as the Turpentine Shed at the London Docks and, although it may not be awe-inspiring to the modern eye, listen to the way Architectural magazine from the period described it.
“On crossing one of the London Docks a short time in the past, we were content to have seen an actual application of Mr. Palmer’s new-fangled roofing. […] Anyone who is observing when passing through it, will be to be amazed (considering it as an enclosure) by its style and simplicity. A small reflection could be able to make them aware of its efficiency and economics.
It is, we consider the lightest and strongest roofing (for the weight) constructed and built by humans from the beginning in the time of Adam. The entire size of the roof was apparent through a close examination (and we even walked over sundry containers of sticky turpentine to serve the purpose of this inspection,) to be not greater than 10 percent of an inch!” [1]
They’re not writing architectural magazines the way they did in the past.
While corrugated material as well as soda bottles are remarkably sturdy however, there’s a way to create materials that are stronger. To see it yourself visit your refrigerator and pull out an egg. Place it in your hand and put your hands around the egg and then squeeze. (Make sure you’re not wearing rings if you try this.) You’ll be amazed by its power. I couldn’t smash the egg, but I put in everything I could. (Seriously I suggest you test the egg to be convinced.)
What is it that makes eggs so durable? The soda cans and corrugated sheets of metal are curving in one direction, but flat in the opposite. This curvature provides them with rigidity, however they could still be shaped into flat sheets from which they originated from.
The eggshells are curved both ways. This is what gives eggs’ strength. Mathematically the doubly curly surfaces possess a non-zero Gaussian curvature. Similar to the orange peel we saw earlier, this implies that they cannot be flattened without breaking or stretching. Gauss’s theory affirms this. To break an egg it first needs to make a dent. If the egg begins to lose its shape it becomes weaker.
The iconic shape of the cooling tower has curvature across both sides. This shape, referred to as hyperboloid, is hyperboloid reduces the amount of material needed for its construction.
Regular chimneys look similar to giant soda bottles as they are strong, however they are also susceptible to bending easily. The hyperboloid-shaped chimney addresses the problem by curving both ways. This double curvature secures the shape in place which gives it extra rigidity which a regular chimney lacks.
Another form that gets its strength from the double curvature would be that of the Pringles potato chips* or, as mathematicians prefer to refer to it as a hyperbolic paraaboloid (say that it is three times speedier).
Nature utilizes the power of this form in a stunningly amazing manner. Mantis shrimps are known for its most powerful punches within the world of animals. It’s a punch so powerful that it evaporates water, producing an shocking waves and an intense light flash. illumination. For the incredibly death punch, the mantis utilizes a hyperbolic paraboloid-shaped spring. The spring is compressed to store this massive force, then releases it with one devastating strike.
Watch the biologist Sheila Patek describe her findings in this incredible phenomenon. or take a look at Destin describe it to you on his amazing YouTube channel Smarter Everyday.
The power that the Pringles shape was a well-known fact by the Spanish-Mexican engineer and architect Felix Candela. Candela is one of Eduardo’s students and built structures which took the hyperbolic parabololoid to new levels (literally).
When you hear concrete, you may imagine boring boxes and slender structures. But Candela could make use of the hyperbolic shape of a paraboloid to create massive structures that showed the astonishing thickness that concrete could provide.
A master of his material He was an innovator in the construction industry and a artist of the structural. His graceful, lightweight structures may seem fragile but they’re extremely strong and built to last.
What does it make that Pringles shape so sturdy? It’s all to do with the way it balances the pulls and pushes. Every structure must be able to support weight and then transfer that weight onto the floor.
They are able to accomplish this in two different ways. There’s compression, in which the weight compresses an object by pushing it backwards. The arch can be described as an example of a structure which is as a result of compression.
There’s also tension, in which the weight pulls on the ends of a structure and stretches it out. You can dangle an end of a chain and each part thereof will become in tension. The hyperbolic parabololoid blends with the very best aspects of each.
The concave U-shaped portion is stretched with tension (shown in black) while the convex arch-shaped portion is compressed (shown by red). By using double curvature, this form achieves a perfect equilibrium between these pull and push forces which allows it to be slim yet extremely strong.
The concept of strength by curvature is a notion that is a driving force in our world and is rooted in geometry. If you take a bite and take a second to examine the area, and take note of the long-lasting legacy of this pizza technique.
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