Tag Archives: geometry

Pointy facts about triangles

The humble triangle has been with us from ancient monuments to the Pink Floyd posters of today. Ancient mathematicians laid the groundwork of geometry by studying triangles, and the shape still terrorizes hapless students to this day.

Image via Piqsels.

The modern word triangle draws its roots from the Latin ‘triangulus’ (‘three-cornered’). It is one of the basic shapes in geometry and the simplest one that can define a plane (a 2-D surface).

What we’re talking about next will refer to Euclidean geometry (the non-weird one).

Types of triangles

As the simplest possible polygon, triangles only sport 3 angles and 3 sides. Their type depends on the relative length of these faces and the size of its angles.

Based their sides, triangles can be classified as:

  • Equilateral: the most aesthetic ones. Equilateral triangles have three equal sides and three equal angles of 60°.
  • Isoscele: triangles that have two sides of equal length, and two equal inside angles. Their name comes from the Greek words for ‘equal’ and ‘leg’.
  • Scalene: with no two sides or inside angles equal, the scalene triangle is the rebel of the lot. Its name comes from the Greek word for ‘unequal’.
Marked sides are of equal length.
Image credits Alexandru Micu / ZME Science.

Based on their inside angles, they’re can also be classified as:

  • Acute: if all internal angles are under 90°.
  • Right-angled: if one angle is equal to 90°.
  • Obtuse: if one angle is greater than 90°.
Image credits Alexandru Micu / ZME Science.

Triangles have some very handy properties. All simple polygons can be broken down into a series of triangles (even triangles). As a rule, the sum of a triangle’s inside angles is always 180°, and each angle is proportional to the side they oppose, with the longest side always opposite their largest angle (the points of triangles are called vertices or vertexes). The sum of the lengths of any two sides of a triangle is always longer than the third side.

This proportionality lies at the root of the field of trigonometry, which deals with the relationship between face length and the internal angles of triangles. Despite its seemingly niche appeal, trigonometry has a huge range of both practical and theoretical applications, ranging from navigation and the tracking of celestial bodies to the analysis of periodic (cyclic) functions, applied science, engineering, and data compression.

History of the triangle

Image via Pixabay.

We don’t know who first discovered — insofar as a shape can be ‘discovered’ — the triangle. But we do know that mathematicians in Ancient Babylon and Egypt had at least a functional understanding of them.

Geometry means ‘measurement of the earth’ and, being the science of shapes and sizes, was likely first developed out of the need to calculate taxes. The hostile environment of Egypt, coupled with the high fertility of the Nile river, led to the implementation of very centralized agriculture practices. Canals and granaries had to be maintained, seeds and tools distributed to ensure enough food was being produced — all this costs money. Government officials used geometry to calculate cultivated surfaces and make sure that the pharaoh’s taxes were being paid in full. Ancient Egyptians also employed astronomy in support of agriculture by allowing for a more accurate prediction of the Nile’s annual flood. Babylonian knowledge of math and geometry were also quite advanced, mostly derived from their astronomical studies, agriculture, and bookkeeping.

Around 2900 BC, the first of Egypt’s pyramids was constructed. These monuments are a remarkable feat of know-how and engineering. In order for a pyramid to actually be a pyramid, the triangles that make up its faces need to slope at exactly the same angle as the others. Since the monuments were built in successive layers starting from the bottom, engineers had to constantly measure and adjust the works with a high degree of accuracy; not easy to do with the tools of the time. Still, we haven’t found any evidence that Egypt’s geometers academically deduced the properties of triangles. They probably worked through observation and solving practical problems. In essence, they were studying applied mathematics.

But, as it tends to happen in history, the Greeks came around and started thinking about things. Thales of Miletus (624–547 BC) is credited with bringing geometry from Egypt into Greece and laying some early groundwork for the study of triangles. Pythagoras, whose famous theorem is still in use, is hailed as the first ‘pure mathematician’ to study geometry by applying abstract mathematical concepts. Later on, Euclid of Alexandria (325–265 BC) wrote a 13-book treatise titled The Elements.

Euclid’s “Elements of Geometrie”, the book’s first translation into English (by Henry Billingsley), published in 1570. You can explore it further here.
Image credits Galileo’s World.

Calling it one of the most important books in Western history wouldn’t be out of place; The Elements still defines most geometric study today (the Euclidean geometric system). What set it apart was that Euclid started with a set of basic data — 23 definitions, 5 general axioms — and a set of 5 postulates (theories). An axiom is a statement that is accepted as self-evident and true without needing proof. Euclid then used the data to provide mathematical proof for his first postulate, which in turn was used to prove his second, and so on (but not his fifth, the ‘parallel postulate‘). His work was so groundbreaking that it underpinned all geometrical study until Carl Friedrich Gauss (1777–1855) and others independently developed non-Euclidean geometry.

The difference between classical (Euclidean) and non-Euclidean geometry largely stems from the parallel postulate. Classical geometry starts from the assumption that parallel lines never meet and that they stay at a constant distance from one another; in non-Euclidean geometry, this isn’t the case. Boiled down, classical geometry works in flat planes, while non-Euclidean geometry operates in curved (elliptical or hyperbolic) planes.

In a non-Euclidean system, triangles don’t abide by the same laws. Imagine drawing a triangle on a flattened balloon — that’s classical geometry. But if you inflate it, the triangle deforms onto a spherical plane, which alters its sides and angle values.

Using Mathematical Functions to Create Stunning Animations

Mathematics is a very powerful tool to create beautiful works of art. By using mathematical formulas, we can create stunning animations. The trigonometric functions (especially sine and cosine) are the most useful mathematical tools to create such animations. Here are a few GIF animations that are made with sine and cosine functions. Each of these animations shows 8,000 moving circles. The sine and cosine functions have been used to determine the center and the radius of the circles.

Credit: Hamid Naderi Yeganeh

Credit: Hamid Naderi Yeganeh

Credit: Hamid Naderi Yeganeh

Credit: Hamid Naderi Yeganeh

Credit: Hamid Naderi Yeganeh

Credit: Hamid Naderi Yeganeh

Credit: Hamid Naderi Yeganeh

Credit: Hamid Naderi Yeganeh

Food-science Sunday : The geometry of a Pringle

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Mathematics is not about equations, numbers, computation or algorithms: It is about Understanding!

There are many ways to understand it – the one that this post is based on is real life visualization.

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The Pringle shape is what is known in mathematics / calculus as a hyperbolic paraboloid.

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Why are Pringles a hyperbolic paraboloid?

The saddle shape allowed for easier stacking of chips. This also minimized the possibility of broken chips during transport.

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Since it is a saddle, there is no predictable way to break it up. This increases the crunchy feeling and hence that weird satisfaction.

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It is relatively more feasible to manufacture the press block compared to other shapes.

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The concave U-shaped part is stretched in tension (shown in black) while the convex arch-shaped part is squeezed in compression (shown in red).

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Through double curvature, this shape strikes a delicate balance between these push and pull forces, allowing it to remain thin yet surprisingly strong.

 

All of this, and also to make some cool ring structures.

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What about Lays?

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Lays is a parabolic cylinder, not as interesting as a Pringles but worth knowing for the sake of completeness.

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You can replicate one with a piece of paper, but you can’t do that with Pringles without cutting the paper and actually adding more paper to it. This makes it more mathematically desirable.

 

Flavor is subjective. Math is irrefutable.

This is what a mathematician had to say:

They’ve got these Lays Stax right next to the Pringles as though they are equivalent. How can they do that?

One is a positive semi-definite quadratic form and the other is an indefinite quadratic form – they’re not even the same definiteness!

 

Sources and more:

Quadratic functions on a banana.

Pringles are NOT potato chips

How it’s made- Stacked potato chips

Best way to hold a pizza slice

Best snack shape

 

Science ABC – how aztecs did the math

aztecThe Aztecs were the dominant civilization in Mexico for several hundred years, when their “reign” was stopped by the Spanish in the early 1500s. An astonishing thing about them (among others) is the fact that they left behind really extensive mathematical writings, intriguing scholars ’til this day.

Two manuscripts in particular have been object to study because they portray land holdings in the Valley of Mexico along with their measurements, using the Aztec numbering system, for purposes of taxation. But now, a mathematician and a geographer have zeroed in on just what methods Aztecs used to measure field surfaces in one of these documents, the Codex Vergara.

The Aztec number system has been deciphred long ago; it is a vigesimal system (using 20 as its base) as opposed to our decimal system. They use dot for 1, a bar for 5, and other symbols for 20 and multiples of 20. The Codex Vergara, painted about 1540, contains schematic drawings and measurements of individual fields. Previous analysis has revealed the fact that they have knowledge of multiplication, division and they even had some principles of geometry.

In a paper that will be published tomorrow in Science the authors show that Aztec surveyors probably used several types of algorithms to calculate area. Some parcels involved simply multiplying length by width. But in other, irregular four-sided lots, they had to come up with different approaches, such as multiplying the average of two opposite sides by an adjacent side. As far as the research shows, they did the math pretty well…especially when it came to taxation!