A fractal is a fragmented geometric shape that is split into several parts, but each of those components is just a smaller-sized copy of the overall form. Many architects apply this mathematical principle to their building designs, like this Los Angeles gas station that recently had a “green” makeover. Everything has been stripped down — including the filling station’s signs, which are subtle symbols — and the mirrored facade beautifies ninety solar panels that power the station. Recycled materials and a plant-covered roof complete the enviro-friendly revamp.
Barcelona’s Endesa Pavillion used mathematical algorithms to alter the cubic building’s geometry, based on solar inclination and the structure’s proposed orientation. Algorithms can be used to create the perfect building for any location with the right computer program. For Endesa, the movement of the sun was tracked on site before an architect from the Institute for Advance Architecture of Catalonia stepped in to complete the picture. The algorithm essentially did all the planning for him, calculating the building’s optimal form for that particular location.
Cube Village
Welcome to Cube Village, built by Dutch architect Piet Blom. His tilted, geometric houses — built on top of a pedestrian bridge to mimic an abstract forest — are split into three levels. The top has windows on every facade and feels like a separate structure entirely.
Magic Square Cathedral
The Sagrada Familia cathedral in Barcelona designed by Antoni Gaudà is a mathematician’s dream. Hyperbolic paraboloid structures are featured throughout. Have you eaten Pringles? Then you definitely know what a Hyperbolic paraboloid structure is. Catenary arches (a geometric curve) abound. The cathedral also contains a Magic Square — an arrangement of numbers that equal the same amount in every column, row, and diagonal. The magic number in Sagrada Familia’s case is 33, which alludes to multiple religious symbols. For example, Jesus performed 33 recorded miracles, and most Christians believe he was crucified at 33 years old in 33 A.D.
Standing 591-feet tall, with 41 floors is London’s skyscraper known as The Gherkin (yes, like the cucumber). The modern tower was carefully constructed with the help of parametric modeling amongst other math-savvy formulas so the architects could predict how to minimize whirlwinds around its base. The design’s tapered top and bulging center maximize ventilation. The building uses half the energy of other towers the same size. Any mathematician would be pleased to claim credit for the building, but architectural firm Foster and Partners might have something to say about that.
Experimental Math-Music Pavilion
Imagine walking up to the Philips Pavilion at the 1958 World’s Fair and seeing this crazy construction of asymmetric hyperbolic paraboloids and steel tension cables. Mind. Blown. This amazing building appeared at the first Expo after World War II, so it was an important moment that allowed its creators to show off the technological progress the world had made since the devastating battle. Philips Electronics Company wanted to create a unique experience for visitors, so they collaborated with an international group of renown architects, artists, and composers to create the experimental space. ArchDaily wrote about the groundbreaking, temporary building, calling it the “first electronic-spatial environment to combine architecture, film, light and music to a total experience made to functions in time and space. It was through these visually inspired concepts that elevated the Philips Pavilion into a complete experience where one could visualize their special movements through a space of sound, light, and time.” Poeme Electronique was one of the works prominently displayed at the time.
Modern Music-Math Home
A classical violinist commissioned an eccentric, $24 million dollar home located on the edge of a Toronto ravine. The curved, elegant structure — which also serves as an incredible concert space for 200 people — was named the Integral House. (Calculus geeks, represent!) The home’s owner Jim Stewart was a calculus professor who wrote textbooks and wanted to incorporate the mathematical sign into the home’s name and design. Undulating glass and wood walls also echo the shape of a violin.
The link between math and architecture goes back to ancient times, when the two disciplines were virtually indistinguishable. Pyramids and temples were some of the earliest examples of mathematical principles at work. Today, math continues to feature prominently in building design. We’re not just talking about mere measurements — though elements like that are integral to architecture. Thanks to modern technology, architects can explore a variety of exciting design options based on complex mathematical languages, allowing them to build groundbreaking forms.
Mobius Strip Temple
We would have made a Mobius Strip in grade school math class, so you should remember that the geometric form is unique in that there is no orientation. A similar twisty shape is applied to the design of Buddhist buildings. The temple is a mound-like shape known as a stupa — similar to a pagoda — and contains a central spire where Buddhists congregate. One architect wanted to modernize it for a soon-to-be built temple in China, and based the updated design on the Mobius Strip — which also happens to symbolize reincarnation.
Tetrahedral-Shaped Church
The tetrahedron is a convex polyhedron with four triangular faces. Basically, it’s a complex pyramid. You’ve seen the same geometric principle used in RPGs, because the dice is shaped the same. Famed architect Walter Netsch applied the concept to the United States Air Force Academy’s Cadet Chapel in Colorado Springs, Colorado. It’s a striking and classic example of modernist architecture, with its row of 17 spires and massive tetrahedron frame that stretches more than 150 feet into the sky. The early 1960’s church cost a whoppings$3.5 million to construct.
Pentagonal, Phyllotactic Greenhouse and Education Center
Cornwall, England’s Eden Project is home to the world’s largest greenhouse, composed of geodesic domes that are made up of hexagonal and pentagonal cells. The social, environmental, and arts/education center is all about green living and considered that in every aspect of their design and programming. Their interactive education center dubbed “The Core” incorporated Fibonacci numbers (a math sequence that also relates to the branching, flowering, or arrangement of things in nature) and phyllotaxis (the arrangement of leaves) in its design.
The relationship between history and Mathematics is reciprocal. History helps mathematics to know about various mathematicians who were pioneers in their field and enriched mathematics by their contributions. History also provides the information about the origin and development of mathematics.
Mathematics helps history in regards to calculation of dates and days etc. of various historical events.
Acquisition of time sense in history is based on the knowledge of mathematics. These two subjects are complementary to each other. Really, speaking, there is a vast difference between the two subjects. Mathematics is generally called a dry subject, while history is supposed to be an interesting subject. But the knowledge of history can make teaching of mathematics quite interesting.
Clara Silvia Roero - Relationships Between History of Mathematics and History of Art
The Narmer Palette, Egyptian, ca. 3000 B.C., an excellent approximation of the catenary curve.
During the course of centuries mathematics has interacted in many ways with culture and human activities, and among these a place of privilege has been reserved for art and architecture.
This paper discusses several examples of the existence of three levels of interaction between mathematics and art: the presence of a mathematical substrate in various archaeological and artistic relics from antiquity, the conscious or unconscious application by artists of mathematical principles whose theories that had not yet been fully developed and, finally, the relationship established by some mathematicians with artists and art theorists that permitted an awareness and acquisition of mathematical knowledge and rules that were then applied to artistic creations. The development of these three levels of interactions between mathematics and art can be a valid aid to the creation of a unified vision of the history of culture of peoples and civilizations, indicating various kinds of influence: technical-practical, theoretical-scientific, mystical-sacred, principles and customs, etc. Indeed, in the wake of a long-term historiographic approach, new research perspectives have emerged recently that have been favourably received by art historians and critics.
About the author Clara Silvia Roero is Full Professor in the Department of Mathematics at the University of Turin, and President of the Italian Society of Historians of Mathematics.
The correct citation for this paper is: Clara Silvia Roero, "Relationships between History of Mathematics and History of Art", pp. 105-110 in Nexus VI: Architecture and Mathematics, eds. Sylvie Duvernoy and Orietta Pedemonte Turin: Kim Williams Books, 2006.
"i" equals the square root of -1, which means that i squared is equal to -1.
Application: Negative numbers don't have square roots. Math had advanced to the point where saying "there is no square root of negative numbers" was holding back a lot of progress.
Solutions of some polynomials have both real solutions that we could use in real life as well as solutions that involved the square root of a negative number, which can be discarded.
Archimedes' constant, or "Pi," is the name given to the ratio of the circumference of a circle to the diameter, but it's actually so much more than that.
Greek mathematician Archimedes is credited with the first theoretical calculation of Pi, which he estimated was between 3 10/71 and 3 1/7 — or 223/71.
Pi is now defined as 3.1415926535... etc
Application: Pi is the key constant in any equation that involves circular or harmonic motion. It's one of the most essential relationships in mathematics.
Euler's number is also known as the exponential growth constant. It is the base for natural logarithms and is found in many areas of mathematics.
Application: In finance, Euler's number is used to determine compound interest, which is extremely vital in understanding the time value of money — the backbone of finance.
Moreover, Euler's number is crucial when describing any decaying relationship - think Carbon 14 dating.
Leonhard Euler, a mathematician with an imagination
Public Domain / Wikimedia
A Swiss mathematician who spent most of his life in Russia, Leonhard Euler is considered the preeminent mathematician of his generation.
Euler was the first to introduce the concept of the function which in and of itself is an immense achievement. That set the stage for all mathematical development since. He was the first to ascribe the letter "e" to mean the base of the natural logarithm, the first to use "i" for the imaginary unit, and he assigned sigma for summation. He introduced Euler's formula, a trigonometric equation, and he developed the Euler's identity, eπi + 1 = 0. His impact on mathematics is profound.
Carl Freidrich Gauss, behind everything we know about statistics
Gauss is considered to be one of history's most influential mathematicians. A German child prodigy, Gauss would later lend his name to an immense amount of discoveries even after his death.
The bell-curved normal distribution is a now central element of modern-day statistics and is sometimes referred to as the Gaussian distribution. Gauss also was interested in the field of differential equations, which are pervasive in modern engineering. He was also central in developing the theorem which established important properties of curvature. He would later co-design the first electromagnetic telegraph in 1833.
German Gottfried Leibniz invented infinitesimal calculus independent of Englishman Sir Issac Newton. His notation is still widely used today.
He was an avid inventor of mechanical calculators and added multiplication and division functions to Pascal's calculator. He refined the binary number system in the late 1600s, enabling the development of digital computers centuries later. A notorious optimist, Leibniz coined the phrase "the best of all possible worlds."
Joseph Lagrange, simplified Newton's work
Unknown Artist, 18th Century
Few mathematicians have made as great a contribution to the field as Lagrange. His legacy is so immense, his is one of 72 names inscribed on the Eiffel Tower and he is buried in the Pantheon, the national tomb for great Frenchmen.
Lagrange essentially created the science of partial differential equations from 1772 to 1785. Today that science is used to model heat, sound, electrodynamics and additional difficult-to-model information. Besides that, he entirely re-formulated and simplified Newton's equations of classical mechanics. Lastly, he also advanced the solution to the three-body problem, one of the trickiest problems in physics.
David Hilbert, besides his immense contribution to functional analysis, may as well be the patron saint of math teachers.
He is one of the founders of proof theory and was a leader in the mathematics field. One of his most important accomplishments was creating, in 1900, a legendary collection of 23 unsolved problems. The problems would go on to set the syllabus for the entire field for the 20th century. In doing so, Hilbert inspired and motivated generations of mathematicians.
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Euclid of Alexandria, prover of math
Mark A. Wilson
Euclid, an ancient Greek mathematician alive during the reign of Ptolemy I in 323-283 BC, was the author of Elements, which served as the primary textbook for mathematics until the dawn of the 19th century. He originated Euclidean Geometry, and while perhaps not demonstrably responsible for the modern era, Euclid was certainly responsible for most of the elementary mathematics that led to it.
Euclid was among the first to formalize mathematical proofs, the primary method of exposition for the discipline.
Chart of the Day in 1786William Playfair / Public Domain
William Playfair, a Scottish engineer, was the founder of graphical statistics. Besides that signature accomplishment, he was at various times in his life a banker, an accountant, a journalist, an economist, and one of the men to storm the Bastille.
It's difficult to overstate his importance. He was the inventor of the line graph, bar chart, and the pie chart. He also pioneered the use of timelines. You're probably familiar with his work.
Pierre-Simon Laplace, pioneer of statistics
Sophie Feytaud, 1841
The marquis de Laplace was pivotal in the development of mathematical astronomy and, most significantly, statistics.
Laplace was one of the first people to propose the existence of black holes. He was one of the central forces behind systematizing probability theory, laying the groundwork for what is now termed Bayesian statistics. He was one of the first to study the speed of sound
This is a information shared in another blog http://123maths.co.uk/blog/maths-is-interesting-here-is-the-proof. From There I am sharing this information.
This story is about a young German boy, solving a maths problem at school, quite a few years ago!!
Here’s the story …
The boy’s name is Carl Friedrich Gauss (1777 – 1855). One day his maths teacher presented a challenging mathematical problem to his class.
The maths problem is to add up all the numbers starting from 1 and ending with 100.
Every student picked up a piece of paper and started to add up the numbers one after another from number 1 onwards.
Within a short space of time, while his fellow students were still struggling, Carl went forward to the teacher and submitted his answer.
That action surprised not only his teacher but the whole class, and his answer was correct!
How did he do that so fast?
Well, he came up with a different way of analysing the problem. Instead of the normal way of adding the numbers 1+2+3 etc, Carl looked at the problem from a different angle.
What he did was to split the range of numbers from 1 to 100 into two equal halves, 1 to 50 and 51 to 100. He noticed that if he flipped the last half to start from 100, and then adding it the two ranges, he will get something stunning.
He discovered that by adding the first pair, 1 + 100, he got an answer of 101. For the second pair, 2 + 99, he again got the same answer 101.
This answer of 101 was still valid for the rest of each number pair addition. And since there were 50 pairs of numbers, the final total is 101 x 50 which gave him an answer of 5050.
The way he perceived and analyzed the mathematical problem surprised everyone.
From this story, you can see that maths is a very interesting subject that tests the human mind. With different approaches, maths solving can achieve a new dimension completely different from convention. This shows that maths can be fun and exciting if we choose it to be!
Mathematics is a methodical application of matter. It is so said because the subject makes a man methodical or systematic. Mathematics makes our life orderly and prevents chaos. Certain qualities that are nurtured by mathematics are power of reasoning, creativity, abstract or spatial thinking, critical thinking, problem-solving ability and even effective communication skills. Mathematics is the cradle of all creations, without which the world cannot move an inch. Be it a cook or a farmer, a carpenter or a mechanic, a shopkeeper or a doctor, an engineer or a scientist, a musician or a magician, everyone needs mathematics in their day-to-day life. Even insects use mathematics in their everyday life for existence.
Snails make their shells, spiders design their webs, and bees build hexagonal combs. There are countless examples of mathematical patterns in nature's fabric. Anyone can be a mathematician if one is given proper guidance and training in the formative period of one's life. A good curriculum of mathematics is helpful in effective teaching and learning of the subject.
Experience says learning mathematics can be made easier and enjoyable if our curriculum includes mathematical activities and games. Maths puzzles and riddles encourage and attract an alert and open-minded attitude among youngsters and help them develop clarity in their thinking. Emphasis should be laid on development of clear concept in mathematics in a child, right from the primary classes.
A teacher fails here, then the child will develop a phobia for the subject as he moves on to the higher classes. For explaining a topic in mathematics, a teacher should take help of pictures, sketches, diagrams and models as far as possible. As it is believed that the process of learning is complete if our sense of hearing is accompanied by our sense of sight. Open-ended questions should be given to the child to answer and he/she should be encouraged to think about the solutions in all possible manners. The child should be appreciated for every correct attempt. And the mistakes must be immediately corrected without any criticism.
The greatest hurdle in the process of learning mathematics is lack of practice. Students should daily work out at least 10 problems from different areas in order to master the concept and develop speed and accuracy in solving a problem. Learning of multiplication-tables should be encouraged in the lower classes.
Another very effective means of spreading the knowledge of mathematics among children is through peer-teaching. Once a child has learned a concept from his teacher, the latter should ask him to explain the same to fellow students. Moreover, in the process all the children will be able to express their doubts on the topic and clear them through discussions in a group.
The present age is one of skill-development and innovations. The more mathematical we are in our approach, the more successful we will be. Mathematics offers rationality to our thoughts. It is a tool in our hands to make our life simpler and easier. Let us realize and appreciate the beauty of the subject and embrace it with all our heart. It is a talent which should be compulsorily honed by all in every walk of life.
Reference: Times of India Newspaper (The writer is a mathematics teacher at Kendriya Vidyalaya, CRPF Amerigog, Guwahati.)
The next topic that needed to be discussed is cone and here comes
A cone is made by rotating a triangle!
The triangle has to be a right-angled triangle, and it gets rotated around one of its two short sides.
The side it rotates around is the axis of the cone.
The pointy end of a cone is called the apex
The flat part is the base
It has a flat base
It has one curved side
It is not a polyhedron as it has a curved surface
We have a good relation of cylinder with cone in respect to volume which we will be discussing with the formulas. This also contains everything surface and volume even depth.
Here comes image of cone taken from Google images and refereed from various websites.
The next shape which we are going to deal is non other than cylinder. The common one which we see in our daily life. We could see cylinder called as Liquified Petroleum Gas. They are said to be cylindrical in shape.
So here follows some of the information and concepts related to cylinder. Here we go
It has a flat base and a flat top
The base is the same as the top, and also in-between
It has one curved side
It is not a polyhedron as it has a curved surface
This is also contains everything surface area, volume and even height it goes and can be discussed later. Here comes the images of cylinder from google images and matter referred from various websites.
Now back with another shape called as torus which deals with ring ball types objects. It is exactly looking like ring ball we used to play with in schools.
This torus has also some important properties which is listed below.
It can be made by revolving a small circle (radius r) along a line made by a bigger circle (radius R).
It has no edges or vertices
It is not a polyhedron
Torus also haves volume and surface area which we can't deny. Here follows the image of torus. These images are taken from google and matter referred from various websites
Now we have Non-Polyhedra shapes which doesn't contain any flat surfaces.I think I have finished fully with polyhedra shapes.
Sphere is similar to a circle but it is a three dimensional figure which means it has volume, depth and everything. The following are some of the common interesting facts of sphere which everyone should know.
It is perfectly symmetrical
All points on the surface are the same distance from the center
It has no edges or vertices (corners)
It has one surface (not a "face" as it isn't flat)
It is not a polyhedron
Sphere will be having a surface area and volume. Their formulas will be defined in the later classes.
Now surface area is nothing but the area covered over the surface and Volume is the capacity to hold things inside it,
Our earth is a example of good sphere as per my consideration. These images are taken from google and also matter refered from various websites.
The Narmer Palette, Egyptian, ca. 3000 B.C., an excellent approximation of the catenary curve.
