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In pursuing whole-person development
The Creative Infographic Design Competition on Applications of Mathematics aims to promote students’ interest in learning mathematics and enhance their awareness of the applications and the positive influence of mathematics in different aspects in real life.
Level
Senior Primary
Award
Silver Prize
Awarded work
Knowing the Great Wall
Theme of the Portfolio
Brief of Work
Do you know the Great Wall in China? This infographic shows different sources of the Great Wall. Including its construction materials, the number of people in each of the six sections of the Great Wall and the number of beacon towers they have. Through this information, people can better understand the Great Wall.
Work
Level
Senior Primary
Award
Bronze Prize
Awarded work
Theme of the Portfolio
Brief of Work
Many architectural structures, including arches and bridges, have used the shape of parabolas throughout history. Rising from the southernmost point of Luis, Qatar, the Katara Towers is an example of parabolic curves being adopted by architects.
On one hand, the parabolically arched design of this tower was inspired by the traditional scimitar swords from the nation seal, this makes it one of the most fascinating architectural designs.
On the other hand, this high-rise tower designed by Kling Consultant has a crescent-like design, making a parabola. A parabola is vertically symmetrical and is like the letter “U” that opens either upward or downward.
Containing symbolic references of the trinity of sand, sea and sky (three elements closely associated with Qatar’s history) and embedding mathematics in the design, the Katara Towers has become a hospitality icon for Qatar.
Work
Level
Senior Primary
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Bronze Prize
Awarded work
Theme of the Portfolio
Brief of Work
This infographic uses progress rings and different shapes (circles and squares) to guide readers through all the steps required to be able to discover how pi is used to relate the height of the pyramid and pyramid’s base perimeter. Thereby demonstrating how advanced mathematics was applied by ancient civilizations in their architecture.
Work
Level
Senior Primary
Award
Bronze Prize
Awarded work
Theme of the Portfolio
Brief of Work
This infographic explains the three main factors which makes a good bridge: (1) Safety, the formula of factor of safety, (2) Appearance, the usage of symmetry and arches used in bridges, and (3) Structure, the reason of why triangle is such a strong shape in bridges and tension and compression applied to bridges with its formula.
Work
Level
Senior Primary
Award
Bronze Prize
Awarded work
Hexagon Power in Architecture
Theme of the Portfolio
Brief of Work
The perimeter-to-area ratio of regular hexagon is compared with those of the other two basic tessellating shapes – equilateral triangle and square. The interlocking boundaries of hexagon tessellation are also highlighted to emphasize the good structural stability. These properties are mentioned with the aim of illustrating why hexagonal design enables the use of the least separating wall materials to divide space or the lightest grid structure to strengthen tall buildings against wind or earthquakes. Regular hexagon has three pairs of parallel sides and looks identical after turning every 60º. Therefore, Hex bolts and nuts allow : 1. good grip for tightening using ordinary wrench while not too easily be rounded off , and 2. reasonably small turning angle for working in tight space.
Work
Level
Junior Secondary
Award
Silver Prize
Awarded work
Theme of the Portfolio
Brief of Work
My artwork shows a town with mathematical structures and busy people doing normal day to day activities. The wide range of buildings in my drawing are inspired by mathematical 3D shapes like cones and cylinders, some of them are also inspired by real architecture like the headquarters of the Bank of China and the Sydney opera house, there are also buildings influenced by mathematical terms I learnt at school like the frustum and Pythagoras theorem. In conclusion, my drawing is to promote mathematics in architecture by fun and interesting buildings.
Work
Level
Junior Secondary
Award
Silver Prize
Awarded work
Theme of the Portfolio
Brief of Work
Mathematics is widely used in our daily life. To start with, I tried to find out its beauty at my school. Therefore, I observed the whole school, even inconspicuous corners.
First of all, I studied the fire resisting door. Arcs and sectors were formed when the door is open and close. I calculated the arc length and area of sector. And I found that the angle was proportional to the arc length and the area of the sector.
Secondly, the semi-cylindrical building of my school was also interesting. When I walked around the stairs every day, I always wondered how big the base of the semi-cylindrical building was. Since the centre of the semi-circle was buried inside the pillar, it was difficult to measure it. However, with the help of the 8 identical windows on the semicircle, it could be estimated by using trigonometric ratios.
Thirdly, the opening of the 8 identical windows was also my concern. On a windy day, I found that they were completely opened. How would the positions of the window hinges be changed if I tried to half close the windows? Then I drew the side view of one of the windows and used Cosine Law to solve the problem.
Finally, the tessellated floor was also fascinated. The basic unit consisted of two squares of different sizes and a rectangle. The pattern of the floor was formed by repeating translating the basic units.
Beside finding the beauty of Mathematics at inconspicuous corners, I have learned drawing skills such as colour matching and composition. In addition, I also learn to express my ideas in clear infographics. Although I felt hard in measuring angles and lengths in the beginning, I am satisfied once I made it! To conclude, I enjoy this research process very much.
Work
Level
Junior Secondary
Award
Silver Prize
Awarded work
The Charm of Architecture - Geodesic Domes
Theme of the Portfolio
Brief of Work
Why are geodesic domes getting more and more popular globally? Geodesic domes are three dimensional structures often used in architectures. They are built up with triangular skeletal struts or flat planes. Sounds very simple right? But how can this transform into a captivating and beneficial architecture that deserves worldwide attention?
Firstly, as you can see from the infographic, geodesic domes are very eye catching, attractive and noticeable. It inevitably stands out from all ordinary rectangular buildings and captures ones heart.
For example, located in Vancouver, Canada, the Telus sphere has been voted the most iconic building in the city.
Secondly, take geodesic domes as hemispheres and compare it with rectangular 3D architectures, letting 2r be the width and r be the height of both shapes respectively. By calculating the surface areas of the hemisphere and cuboid, as seen in the infographic, the cuboid have almost twice the surface area as the hemisphere. This indicates that the building cost of geodesic domes are way cheaper if the same constructing material is used.
Moreover, geodesic domes are self-supporting and very stable, stabilised by the forces of gravity. In other words, pillars and intermediary columns are not required. Hence ending up in more free spaces below, enclosing a maximum space with a minimum of inner volume.
Last but not least, the natures of geodesic domes are friendly to the environment. The spherical designs results in effective air circulation in both summer and winter. They are also naturally more energy efficient when compared to standard architectures due to unobstructed interior and exterior air flow. The relatively smaller surface areas makes these buildings less susceptible to temperature changes as well.
All in all, geodesic domes are winners in terms of appearance, cost and energy saving. Let’s discover more geodesic domes out there!
Work
Level
Junior Secondary
Award
Bronze Prize
Awarded work
Theme of the Portfolio
Brief of Work
I chose this ‘The Center of Gravity of Buildings’ topic because I think the center of gravity that is involved in buildings are pretty interesting. The center of gravity keeps the architecture sturdy and not fall down. Moreover, I think this topic is pretty challenging because the amount of data that can be used in this topic is very small so it will be harder to do the full infographic. The center of gravity is also very common in daily life, like when we are trying to balance or piling thing up without letting them drop to the ground etc. Next, my contents in this infographic is introducing the center of gravity and telling inner structures of buildings. I can write a lot of information in this infographic since my topic is suitable for that. My design is kind of a statistics-based one. It has more facts than other graphs.
Work
Level
Junior Secondary
Award
Bronze Prize
Awarded work
Mathematics in Architecture−Demonstrating Mathematics in the Context of Architecture
Theme of the Portfolio
Brief of Work
The topic of the infographic is Mathematics in Architecture. In my opinion, we use math in architecture on a daily basis to solve problems. We use it to achieve both functional and aesthetic advantages. As you will see from some of the examples below, the application of mathematical principles can result in beautiful and long-lasting architecture which has passed the test of time. Mathematics has various roles in architecture. In the ancient Egypt in 300 B.C., architects used the Golden Ratio to design proportions in buildings that look pleasing to the human eyes and feel balanced. For example, the Parthenon is a temple built on the Acropolis in the 5th century BC for the Greek goddess Athena. It appears to use golden ratio in some aspects of its design to achieve beauty and balance its design.
The golden ratio is an irrational number where a line can be divided such that the long segment divided by the short segment is approximately 1.618. Also, the sum of the lengths of the segments divided by the longer segment is approximately 1.618.
For the Great Pyramid of Khufu if we take a cross-section through a pyramid we get a triangle which was called Egyptian Triangle. The strength of a triangle derives from its shape, which spreads forces equally between its three sides. Triangles are stable, as they are inherently rigid, the three sides mutually reinforcing each other. That’s why the pyramid lasts long.
The above examples show that architecture and mathematics seem to have few obvious connections, but despite the apparent differences, the distance between the profession of architecture and the discipline of mathematics, and between an object of design and a subject of study is far less than many would assume.
Work
Level
Junior Secondary
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Bronze Prize
Awarded work
Mathematics in Architecture
Theme of the Portfolio
Brief of Work
The topic of the infographic is Mathematics in Architecture, which consists of two parts: “How Mathematics makes buildings more stable” and “How Mathematics makes buildings more beautiful in shape”.
The first part is about the use of triangles. Being the strongest shape among all geometric shapes, triangles are commonly used in buildings. For example, trusses are structures commonly seen in bridges. The main reason is because they are rigid. Unlike other polygonal structures, they would not change their shapes even under force. When a force is applied, it is distributed to the two sides and the remaining side is stretched. Therefore, the shape is still stable. However, other polygons are not that rigid and would be easily compressed when force is applied (for example, rectangles would be compressed into parallelograms).
The second part is about how Golden Ratio were found in architectures. The Golden Ratio is an irrational number approximately equal to 1.618 and satisfies the equation Φ² – Φ – 1 = 0. The Golden Ratio is considered beautiful and can be seen in a number of buildings like the Great Pyramid of Giza. The Pyramid has a base of 230.4 m and a height of 146.5 m, which has a height-to-base ratio of 0.636 and is in the shape of a Golden Triangle (a Golden Triangle is a right-angled triangle with base-to-height-to-hypotenuse ratio equals 1 : √Φ : Φ, such that the three side lengths would satisfy the equation Φ² – Φ – 1 = 0 due to the Pythagoras’ Theorem).
Furthermore, the Golden Ratio is related to the Fibonacci sequence, in which the first two terms are 1 and starting from the third term each term equals the sum of the two previous terms. The ratio between every two consecutive terms approaches the Golden Ratio. The numbers in the sequence form the Fibonacci Spiral, which can be seen in the nature.
Work
Level
Junior Secondary
Award
Bronze Prize
Awarded work
Theme of the Portfolio
Brief of Work
Mathematics concepts and ideas are everywhere, as well as at the corners of my school. They can be found at doors, windows, gardens and even the floor plan of my school.
One of the corners is the classroom door. There is a round-shaped window on it. When I tried to measure the diameter of the circle, I had some difficulties to locate the centre. To solve the problem, I got a few measurement by sliding my measuring tape in a parallel direction. The maximum value of the measurements was the best approximation value of the diameter of the circle.
Besides, I also wanted to study the angles between the door and the wall. It was amazing to find out that, even though arc lengths and areas of sectors were proportional to the angles, the areas of triangles were not! (This could be explained by the graph of a sine curve!)
The floor plan of my school was also inspirational. The floor plan consisted of 6 classrooms, a corridor and the Spiritual Garden. I used rate and ratio to draw the floor plan. Also, I used a benchmark (i.e. the length of a piano) to estimate the length of the corridor. For The spiritual garden, it is a rectangular area with square platforms with triangular spaces by two sides. If the school wanted to buy some pebbles to decorate the triangular spaces, decomposition-composition strategy could be used to estimate the total number of pebbles.
Work
Level
Junior Secondary
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Bronze Prize
Awarded work
Stairs
Theme of the Portfolio
Brief of Work
Staircases are one of the many things we come across almost every day – yet, do you know how they are constructed to be the sturdy, sustainable and precise infrastructure we all know? The infographic explains how mathematics is used in architecture – namely stairs. By calculating the exact rising (height) and going (width) of the steps, they are all identical and equal, with the same rising and going. We can also calculate the angle of the stairs by dividing the total run by the total rise and using the quotient to calculate its inverse tangent. Studies show that an angle of under 41° is recommended for stairs. This ensures that the stairs are easy to climb and not too steep, so we don’t waste our energy and save it for other uses. The infographic also includes a useful table that states the recommended maximum and minimum rising and giving the height of the steps. The goal of the infographic is that people can design the stairs of their dreams with guidelines that are easy to understand!
Work
Level
Junior Secondary
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Bronze Prize
Awarded work
Why are triangles often used in architecture?
Theme of the Portfolio
Brief of Work
Congruence is a topic learned in junior form mathematics. Most students only recognise SSS as a condition of congruence, many of them aren’t aware that it is related to the rigid shape of a triangle.
Therefore, I typed it on a drawing of the memo paper. In contrast with the simple and plain background, it will immediately stand out and capture the attention of the readers.
In order to visualize how solids change their shape deform when force is applied, I drew the ‘before and after’ of the 3D solids. I pointed out triangles are the best at taking force. The data is used from a real experiment shared by Participant J0306 of the California State Science Fair, who shared his findings online.
The next step is to apply it back to architecture. With the help of a bridge simulator, I found out what would happen if we use different shapes to build bridges. If I used words to express it, it would make the layout messy, affecting the reading experience of the readers. Because of this, I used pictures to again, help the readers better understand the situation. Lastly, I traced the shapes of famous bridges around the world and highlighted the parts that used triangles, linking the infographic back to this year’s topic ‘Mathematics In Architecture’.
We can bring the flat 2D shapes on our math books to real life, apply them to architecture and help people build safer and cheaper structures. One of the major comments that most people have on mathematics is that it is useless and won’t be applied in real life. I hope my infographic will prove to them that mathematics can be applied in real life and is beneficial to society.
Work
Level
Junior Secondary
Award
Bronze Prize
Awarded work
Theme of the Portfolio
Brief of Work
This infographic introduces the biggest Egyptian pyramid: The Great Pyramid of Giza. Since the main colour tone of the Egypt deserts and the pyramid is orange, a bright orange is used as the main colour theme of the whole graph. The infographic includes some basic information of the pyramid such as the materials used to make the pyramid as well as the height and base of it, which are used to calculate the area and the volume later on. At last, there is a part where a real life photo the pyramid is pasted and some geometric features (3D shapes) are introduced in the graph.
Work
Level
Senior Secondary
Award
Bronze Prize
Awarded work
Mathematics in Architecture of Forbidden City
Theme of the Portfolio
Brief of Work
The Palace of Celestial Purity, located within the Forbidden City in Dongcheng, China, is an architectural classic and has an impressive design inspired by the traditional Pagoda style. This style is characterized by its symmetrical roofs, a feature that is executed in the Palace of Celestial Purity, which was constructed in the early 15th century. To further explore the symmetrical design of the Pagoda-style roof, a quadratic function f(x) = ax2 + bx + c, where a ≠ 0 and b and c are real numbers, was used to demonstrate the roof’s symmetry. In order to invert this quadratic function, f(x) = f(-x) for all values of x, so that the function is symmetric along the y-axis. This involves horizontally inverting the function along the y-axis, which results in the equation remaining the same except for the sign of the constant term “c” becoming negative in a quadratic function and “a” becoming negative in a linear function. To demonstrate the symmetry of the roof along the y-axis, 2 quadratic functions and 3 linear functions were used on one side of the roof, which were then inverted to create a symmetric shape that should perfectly fit the roof. For example, the equation f(x) = 0.25x2 + 2.13x – 6.54 was used to demonstrate the first slope of the roof. By inverting this function, the equation becomes f(x) = 0.25x2 – 2.13x + 6.54, as demonstrated in the accompanying infographic. To confirm the symmetry of the roof, another function was created and compared with the inverted function. By matching the equation, it was concluded that the architecture behind the roof was accurately symmetrical. This demonstration effectively showcases the importance of mathematical principles in architecture, and how they can be used to create stunning and symmetrical designs that are visually appealing.
Work
Level
Senior Secondary
Award
Bronze Prize
Awarded work
Theme of the Portfolio
Brief of Work
The infographic design of the Parthenon comprises a collection of imagery, data, minimal text as well as engaging visuals that conveys an overview of the Parthenon which was built in the 5th century BCE, also known a’s the Golden age of Pericles. It is well known for its harmonic proportions, precise construction, and lifelike sculptures. A 38 feet tall statue of Athena, the Greek goddess of wisdom and warfare, once stood inside the Parthenon. The Parthenon has a seemingly perfect structure. The Golden Ratio, in which the temple was built, is a ratio between two numbers that equals approximately 1.618. The Doric columns measure 1.9 m in diameter and are 10.4 m high. The Parthenon had 46 outer columns and 23 inner columns. The Parthenon’s floor area is measured 73 by 34 m, which is 2484 m2.
Work
Level
Senior Secondary
Award
Bronze Prize
Awarded work
Theme of the Portfolio
Brief of Work
Tessellation is very common in our daily lives—from tiles to architecture and I could see it on my way to school every day. Seeing it all the time makes me curious about the reason why tessellation is so commonly used. That’s why I chose it as the theme of my infographic.
Tessellation is covering a flat surface using one or more geometric shapes, with no overlaps or gaps. A regular tessellation using equilateral triangles could be usually seen as triangles are the strongest shape among all polygons. Any weight placed on it is evenly distributed on all 3 sides.
To check whether a polygon or more than 2 polygons are able to form tessellation, a formula proposed by Grünbau and Shephard could be used. For instance, equilateral triangles and squares could give rise to a semi-regular tessellation in a “33344” or “33434” pattern. It is really interesting to check and try to tessellate polygons by myself.
With a spatial tessellation design, the load distribution could be more reasonable in structures, and the damage of a single component would not lead to the collapse of the whole building.
Tessellation facilitates constructing the geometries by simply repeating regular or non-regular shapes, which gives designers the freedom to create more complex forms using a simple design approach. La Seine Musicale in Paris and the Bank of China Tower in Hong Kong are some examples of the use of tessellation.
Before doing this project, I had never realized that Mathematics is all around us. Mathematics in daily life differs from the one in textbooks. It is more understandable, which arouses my interests in Mathematics. In order to discover more daily Mathematics, I would pay attention to everything on my way, instead of just focusing on my mobile phone.
Work
Level
Senior Secondary
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Bronze Prize
Awarded work
When Architecture Meets Mathematics
Theme of the Portfolio
Brief of Work
The topic of the infographic is “When Bridges Meet Mathematics”. It consists of three parts that explain how mathematics applies in the construction of bridges and how bridges can be stable and able to withstand strong forces.
Firstly, the parabola concept is involved in the design of a bridge. Parabolas are often found in architecture, especially in the cables of suspension bridges. Since the stresses on the cables as the bridge is suspended from the top of the towers are efficiently distributed along a parabola, the bridge can remain stable against the forces that act against it. To build a parabola, engineers should first find out the starting and ending points, then apply them in the quadratic equation: f(x)= a(x-h)^2+k.
Secondly, the infographic is about geometry. In order to design a bridge with the best angles and make structures as strong as possible, engineers use geometry to find out the most suitable shape, size, position, etc. The most recognized geometric shape is a triangle since it provides strength and stability. When a force is applied to the vertex of a triangle, the two lateral sides squeeze together and the bottom side pulls apart. Each side experiences only one force at a time. Hence, the triangle does not break. Today, triangles are the most common shapes used in construction.
Thirdly, symmetry is also highly applied in the design of bridges. Bridges have reflectional symmetry so that each side of the bridge looks the same and thus equal force will be put on each side.
In conclusion, the above shows that several Mathematical concepts are used to construct a bridge so as to make it stable. It is believed that Mathematical concepts are not only used to build bridges but also other architectures. Thus, Mathematics apparently is a crucial partner in construction.
Work
Level
Senior Secondary
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Bronze Prize
Awarded work
Mathematics— the best tool to measure energy efficiency of architectural design
Theme of the Portfolio
Brief of Work
In our daily life, we usually see various buildings in different shape and materials. Do you know the reason? It is because one of the considerations is to improve the energy efficiency and it can be proved by mathematics.
The first factor is the shape of building. The energy efficiency measurement can be calculated by Surface Area to Volume Ratio. A building with a small surface area compared to its volume will be more energy efficient. Three types of buildings, quadrangular pyramid, cuboid and cylinder, are used to compare energy-efficiency performance and those buildings are commonly found as examples. From the calculation result, the building in quadrangular pyramid shape reveal the best energy-efficient performance.
The second factor is the material use. The energy efficiency of the material can be measured by U-value. It is used to measure how well or how badly a component transmits heat from the inside to the outside. The slower or more difficult it is for heat to transfer through the component, the lower the U-value. This means that we are looking for a lower U-value. A bar chart of U-values of different materials is found. From the chart, we can see that timber is the best for energy efficiency. In conclusion, building with quadrangular shape and made with timber is the most energy efficient.
However, in real life, the architect also needs to consider the applicability, safety and economic reason while designing the building. Therefore, it is rare to see buildings made with the best shape and material. Without a doubt, mathematics is extremely important for architecture. If Mathematics didn’t exist, people would have problems to improve building’s energy efficiency!
Work
Level
Senior Secondary
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Bronze Prize
Awarded work
Pyramids- with the magic of mathematics
Theme of the Portfolio
Brief of Work
A pyramid is a structure whose outer surfaces are triangular and converge to a vertex at the top. Its base is usually triangular or rectangular. It is often found in Egypt. Meanwhile, a lot of mathematical theories are behind the construction of pyramids.
First, the cross-sections of pyramids on each side of them are triangles. These triangles are isosceles triangles so we can actually cut them into two right-angled triangles. Many mathematical theories work on right-angled triangles, such as Pythagoras’ Theorem. It means that the square of the length of the two sides of triangle is equal to the square of the length of hypotenuse, known as a2+b2=c2
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Also, some trigonometry can be applied to find out the interior angles of the triangle. sin θ = opposite side/hypotenuse, cos θ = adjacent side/hypotenuse and tan θ = opposite side/adjacent side. Using these methods, we can easily find out the inclination of a certain pyramid, and hence find the height, width, length of a pyramid without accurate measuring.
Secondly, the construction of pyramid may involve the use of golden ratio. It is said that paintings or architectures that has made use of the golden ratio are aesthetically pleasing. The golden ratio is denoted as phi φ, which is equal to (1+√5) /2 , around 1.618. Why this ratio is so special? It is because it is found in many natural things. For example, the pattern of a sunflower, snail, etc. A triangle of golden ratio is called the Kepler triangle. Ratio between its base, side and hypotenuse is 1: √φ : φ, in which the Pythagoras’ theorem is applicable in this triangle. Knowing these facts, we can know understand how pyramids are built and why they can be so magnificent.
Work
Level
Senior Secondary
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Bronze Prize
Awarded work
Hyperboloid and Hyperbola in Architecture
Theme of the Portfolio
Brief of Work
Nuclear energy is an important energy source of the world. Nowadays, over 400 nuclear reactors in more than 50 countries provide about 10% of the world’s electricity. I will focus on the cooling towers that are usually found close to the nuclear plants. Cooling towers are essential facilities of nuclear plants for removing large amount of heat and energy in the nuclear reactions and releasing them into the atmosphere in form of water stream. I attempt to illustrate the applications of mathematics by revealing the uses of hyperboloid and hyperbola in my infographic work beyond the equations and formulas.
Cooling towers are built with hyperboloid shapes which can be formed by rotating a hyperbola about its axis and a hyperbola is an open curve with two branches that is formed from the intersection of a plane with both halves of a double cone. Mathematically, hyperbola can be regarded as the locus of a moving point whose difference in distances between two fixed points (called foci) is a fixed constant.
The advantages of the hyperboloid shape of cooling tower includes but not limited to reducing resource consumption and the environmental impact during the nuclear power production. Not only does the curved shape of hyperbola of the hyperboloid provides a strong and powerful structure, but it also has many practical functions. The broad base of the tower facilitates evaporation due to its relatively greater area. The narrow middle part can speed up the air flow and thus improves the cooling efficiency. The tower is widening from the middle to the top in shape to allow mixing of the moisture laden air into the atmosphere.
Work
Level
Senior Secondary
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Bronze Prize
Awarded work
Unexpected Footprints of Binomial Coefficients in Bridge Triangles
Theme of the Portfolio
Brief of Work
Ting Kau Bridge is the longest 3-tower cable-stayed bridge in the world. The 1177m-long cable-stayed bridge is supported by three single-legged towers and 4 planes of stay cables radiating from the top of the towers. This infographic work was inspired by the many triangles in the spectacular side view of the Ting Kau Bridge.
Stewart’s formula relates the lengths of three sides and a Cevian of a triangle which is a line intersecting one of the vertices and the side opposite to that vertex. If the Cevian bisects the opposite site, then Cevian is a median. Referring to the triangle in my infographic work, in DPQR, PS is a median and
PQ2 + PR2 = 2PS2 + 2QS2
If we extend the base to another point T outside the triangle so that RT = QS = SR, then PR is a median of DPST and hence
PS2 + PT2 = 2PR2 + 2SR2.
Since QS = SR, by eliminating QS2 and SR2 in the above two equations, we can obtain
PQ2 + PR2 – PS2 – PT2 = 2PS2 – 2PR2
PQ2 – 3PS2 + 3PR2 – PT2 = 0.
If we repeat further extend QT to U such that UT = RT = QS = SR, using similar method, we can obtain
PQ2 – 4PS2 + 6PR2 – 4PT2 + PU2 = 0.
As I work out more equations, one thing to catch my attention is the coefficients of the terms in these equations. They are binomial coefficients in Pascal’s Triangle (the coefficients of the terms in non-negative integral powers of binomial expansions). This is the first time I notice the footprints of binomial coefficients in geometry. This is the reason why I develop this infographic work to share my joyful experience of seeing the connection between mathematics and the world with others.
Work
Level
Senior Secondary
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Bronze Prize
Awarded work
U-shaped Curves in Architecture
Theme of the Portfolio
Brief of Work
Have you ever wondered about the U-shaped curves found in many architectures such as bridges, arches, cables or even hanging chains? Most people would think these U-shaped curves are parabolas, just like the path traced out by a projectile and so do I. Genius scientist Galileo also thought the curve of a hanging chain was a parabola and he later found out this was a wrong idea. Joachim Jungius later proved that the curve found in a hanging chain is not a parabola mathematically.
In fact, this kind of curve is called a catenary which is the shape of a chain or cable with its own weight supported by the two ends under normal gravitational assumption. On the other hand, a projectile has no external supports of its weight and just falls freely under the effect of the gravity and probably its initial speed. Graphically, if we put a parabola and a catenary together, we can see that the catenary grows more slowly at the lowest point comparing with the parabola and the growth rate at other points of the catenary is much bigger than the same corresponding points on the parabola. Mathematically, the equation of a typical catenary is 
where a is the distance of the lowest point from the x-axis while the equation of a typical parabola is y = ax2 + bx + c.
Work
Level
Senior Secondary
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Bronze Prize
Awarded work
A tale of two buildings in Architecture Photography
Theme of the Portfolio
Brief of Work
I share with readers the tips for capturing two architectures in the same photograph so that neither one of them becomes too big nor too small even if they are not next to each other. We describe how we can capture two important landmarks located in Central, namely, International Financial Centre (IFC) and Bank of China Tower (BOCT) and offer a solution of this architecture photography problem in two steps: What photograph-taking spots we should choose and where to locate these suitable spots out.
To find a suitable photograph-taking spot, we should first find the height ratio of the two buildings (the height of IFC to that of BOCT is 1.31308:1). The photograph-taking spot should be chosen with the ratio of the distance from the IFC to that from BOCT approximately equal to 1.31308:1 so that the two right-angled triangles are similar to each other and the angles between the horizontal and the line joining the spot and the top of the buildings are approximately equal. These two equal angles will be preserved when the light enters the lens of the camera and fall on the focal plane of the lens. As a result, the heights of the two buildings look equal in the photograph.
To locate these photograph-taking spots, we need a geometric fact: if a moving point maintains a fixed distance ratio from two fixed points, its locus is a circle known as an Apollonius Circle. This offers us an easy way to locate these spots. We need to locate a point on this circle (with the ratio of the distances of the two buildings equal to the height ratio) with a clear view of the two buildings and the images of the two buildings will look approximately equal in height in the photograph.
Work
Level
Senior Secondary
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Bronze Prize
Awarded work
Theme of the Portfolio
Brief of Work
This infographic aims to show the usage of mathematics in architecture. By the typical example of the leaning tower of Pisa, the importance of maths is shown. It helps us figure out its maximum tilt of standing so that the risk level can be estimated, to prevent the exposure to its collapse.
Work
Last revision date : 24 October 2024
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