#exploremaths Examples of the Golden Ratio in Nature and Architecture (By Michelle So)

Golden Ratio – Nature

1. Pinecones- The seed pods on a pinecone are in golden
ratio as each pair of spirals are in the cone, spiral upwards in different
directions, taking steps which will match a pair of consecutive Fibonacci
sequence. 

2. Tree Branches – the golden ration is shown through the
way tree branches split. When the tree grows old enough to grow branches, it
will split into two, then one of the two will split again, while the other is
to remain dormant.

3. Spiral Galaxies – the shape of the galaxies is
following the golden ratio as each of the spiral arms has a logarithmic spiral
of about 12 degrees. This relates to the golden ratio as it logarithmic spirals
are golden ratio spirals which appear in nature.

 

Golden Ratio – Architecture

1. Mobius Strip Temple – it is a Buddhist temple made out of
unique geometric shapes that has no orientation.

2. Tetrehedral Shaped Church – a complex pyramid in the
shape of a Tetrehedral, which is a convex polyhedron with four triangular
faces.

3. A mathematically- inclined cucumber in the sky – it is
a building in a shape of a cucumber with 41 floors and is 591 feet tall. In order
to create this tower, many mathematical equations and formulas were used.

The golden ratio can be seen almost every where around the world, if the golden ratio didn’t exist, it would be affecting many both in nature and architecture.

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The Prescence of the Golden Ratio and the Golden Spiral throughout the World

#exploremaths

When looked at from the front, the shape of the Parthenon appears to resemble a golden rectangle. This is interesting as construction of it had completed in 438 B.C.E but it wasn’t documented by the Greeks until 300 B.C.E. Many theories for this include that, as it is present in nature, the golden ratio is aesthatically pleasing to the human eye.

 

As seen above, the Taj Mahal, built in 1648 by Ustad Ahmad Lahore for Mughal Emperor Shahjahan and his wife, is also known to form a golden rectangle from a frontal view

A more modern example of the golden ratio in architecture, the UN Building has 4 levels with clear glass, forming 3 golden rectangles.

​Pinecones display many golden spirals from the centre of seed outwards.

It is surprising, personally, to see that the golden ratio is present in inanimate, abiotic objects. The Milky Way, our home galaxy, has several golden spirals extending from the centre. 

​Even at a miniscule level, the golden ratio is known to exist. Each DNA molecule measures 34 angstroms long by 21 angstroms wide and as learnt in class, the higher the consecutive fibonacci numbers are, the closer and closer the ratio of these 2 numbers gets to phi.

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Exploring Mathematics: The Golden Raito in architecture and nature ( By Kevin Xu)

Exploring Mathematics: The Golden Ratio in architecture
and nature

The Golden Ratio in architecture can be found in many famous
buildings around the world. These buildings include:

·        
The Parthenon in Greece

·        
Notre Dame in Paris

·        
Taj Mahal in India

The Parthenon in Greece uses the Golden Ratio in the height
of the columns, the dividing line of the root support beam and the width of the
columns. The golden ratio of the columns is the structural beam on top of the
columns which is in proportion of the golden ratio to the height of the
columns. The dividing line of the root support beam is in proportion to the
golden ratio as the structural beam on top of the columns has a horizontal
dividing line. The width of the columns is in proportion formed by the distance
from the center line of the columns to the exterior columns.

The Notre Dame in Paris have three major architecture lines.
They are commonly known as the Red, Blue and green lines. The red line is the
vertical line height of the base at ground level in ration with the top of the
first and second level. The blue line is the vertical height of the base at
ground level proportion to the top of the second and third floor. The green
line is horizontal width outside of the left top section proportion the inside
of the top right section and the outside of the top right section.

The Taj Mahal in India is another example of the golden
ratio. It itself displays the ratio proportion in the width of the grand
central arch to its actual width and also the height of the windows inside the
arch to the height of the main section under the domes.

The Golden Ratio in nature can be found in various types of
plants. Examples of them are:

·        
Romanesque Broccoli

·        
Hurricanes

·        
Sunflowers                        

The Romanesque Broccoli has always had a spiral starting
from the center and all the florets are formed around the spiral thus creating
the Fibonacci spiral which consist of a series of smaller arcs with a radii
that will imitate the Fibonacci series.

A Hurricane is also an
example of the golden ratio as it has the golden ratio spiral formed inside the
hurricane from the center.

The sunflower is the
most common example of the golden ratio. The spirals in the sunflower use the Fibonacci
series to create their seed heads.Those seeds should come out golden!

 

 

 #exploremaths

5 Unique Examples of the Golden Ratio in Life (Josh Luong)

1. Research 2-3 examples (architecture)

The Taj Mahal: The Taj Mahal is located in India and is a
Mausoleum made from white marble. The design of the building is shown to use
the golden ratio in the width of the grand central arch and the heights of the
windows inside the arch shown in the picture below.

Notre Dame: The Notre Dame is located in Paris, France and
is a historic Catholic cathedral. The golden ratio is shown in the height of
the building and also seen in the structure of the columns at the top. Each of
the 3 golden ratio used is difference in proportion to each other.

The Parthenon: The Parthenon is
located in Athens, Greece and was a temple, which was dedicated to a Greek
Goddess, Athena. Even if the Parthenon was constructed with the golden ratio
proportions, people argue whether if the ratio was intended or whether it a
mere coincidence.  The golden ratio
can be seen across all the parallel pillars making 1 golden rectangle.

2. Research 2-3 examples (nature)

Spiral Galaxies: The Golden ratio has no limits and
can see seen in the spiral of the galaxies, which include our milky way. 

The Human Face: Even if
everyone’s face is unique, the average human face tends to be close to the
golden ratio. The eyes, lips and forehead all resemble the golden ratio.  Apparently the closer our faces look
like “phi” the more attractive we look. Very faceinating!

#exploremaths

Examples of the Golden Ratio in nature and architecture

1) Golden rectangles in architecture
Notre Dame

The Notre Dame in Paris expresses the Golden ratio in the design of the structure. The building is quite asymmetrical which makes it difficult measure the ratio.

The United Nations Secretaria

the United Nations building conveys the idea of the Golden ratio and was completed in 1952. the structure was constructed by Oscar Niemeyer and Le Corbusier.

Taj Mahal

The Taj Mahal was completed in 1648 and was recognise as the Divine Proportion. This was because of the use of the Golden Ratio thorugh out the many features of the building. 

2) Golden spiral in nature

Spiral Galaxyies

In 1925, astronomers realized that the spiral in the milky way was in respect of the Golden Ratio. 

Hurricanes 

Pine cones 

The spiral arrangement of the seed pods on a pine cones portrays the Golden Spiral one after another. 

#exploremaths 

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Examples of Golden Rectangle #exploremaths

#exploremaths

1. Research 2-3 examples (architecture):

The Parthenon uses the golden ratio to make up height and length of the building, including the columns that hold it up. Although it is assumed to use the golden rectangle’s proportions, it is debated that The Parthenon was not built based upon the golden rectangle or spiral.

The Notre Dame is known to have a number of golden ratio proportions in it’s main designs. The bottom is the biggest golden rectangle, and then the sizes gradually become smaller golden rectangles. The Notre Dame is a very clear example of the use of the golden rectangle.

The UN Secretariat Building, like The Parthenon, is also debated whether or not the golden rectangle was used. Although the clear rectangular shape of the building makes it highly likely to be built based on the golden ratio, it is also assumed to have used the proportions of the human body (based on Leonardo da Vinci’s Vitruvian Man).

2. Research 2-3 examples (nature):

The human body has a divine proportion. the entire height of a person (from head to toes) can have golden ratios. The golden section of the entire height would reach from the head to the fingertips. Then the golden ratio of that would be from the head to the navel etc. and it keeps going.

Sea shells also have the same dimensions, measuring from the outermost spiral growth, and gradually getter smaller as it closes into the innermost spiral growth.

In a dolphin’s body and fins, they also carry a similar dimension to the previous two examples. The length from the dolphin’s tail to the head corresponds to the section’s of the dolphin’s tail.

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The Presence of the Golden Ratio and the Golden Spiral throughout the World

#exploremaths

When looked at from the front, the shape of the Parthenon appears to resemble a golden rectangle. This is interesting as construction of it had completed in 438 B.C.E but it wasn’t documented by the Greeks until 300 B.C.E. Many theories for this include that, as it is present in nature, the golden ratio is aesthatically pleasing to the human eye.

 

As seen above, the Taj Mahal is also known to form a golden rectangle.

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My not-so-new workspace

12 months ago I moved to a new school and had the fun task of setting up my desk from scratch (again). It was very bare back then – this is how it looks now:

2015/01/img_3483.jpg

As you can see, the whole thing is rather more filled out than before. My shelves are now looking a little healthier (or they may need to lose a bit of weight, depending on your preference) and my screens seem to have grown and multiplied. (Seriously, trying to do any kind of timetabling on a single monitor? Practically an invitation to madness.)

I’m feeling a lot more settled and directed heading into 2015 – I guess that’s to be expected since this is my second year in the role. Nonetheless, there is plenty of change afoot – a couple of new staff in my faculty, and I’m very excited about the energy they’re going to bring to the table. One thing’s for sure: there are interesting times ahead!

 

Teaching outside the classroom

Hey Eddie, I got offered a full time job next year. Pretty excited but I want to be an active member within the school, rather than just doing the teaching component. So for example I want to participate in extra curricular activities within the school. But i was wondering if that is something a first year out teacher should do, considering the workload? What are your thoughts on that?

What a fantastic question. It comes from a really good heart that wants to get involved but also wants to be realistic about its own limits (and believe me, in your first year of teaching you will definitely confront your limits!). This is something I personally feel very strongly about, and I’m glad you’re thinking about very consciously rather than just falling into the path of least resistance.

Full disclosure: every year that I’ve been a teacher (including my first), I’ve been deeply involved in activities outside the classroom. That probably gives you an indication of the kind of advice I’m about to give. But now what’s important is that I justify it!

To answer your question, I need to pose a question back to you first. And that question is: what is a teacher’s role? What is a teacher actually supposed to do? You’ll note that I did not ask, “What is a teacher’s job?” or “What is a teacher paid to do?” I don’t want to know what the minimum expectations are; I want to know what it actually means to be a teacher, what it should mean, what it ought to mean. As a teacher, who are you? Why do teachers exist?

Your answer to my question will define the answer to your own question. I’ll show you how I answer each one in turn.

As a teacher, who am I? Why do I exist? I am a role model, an instructor, a mediator, a mentor, and a coach. I exist to do everything I can to bring people through the journey from being a child to being an adult. That includes standing in front of a classroom and explaining things – a lot of it – but it includes so much more. It includes teaching children how to relate to their peers – and those who aren’t their peers. It includes being an example of character and integrity. It includes forcing students to challenge their own limits even when they don’t want to – and proving to them that they are capable of more than they believe they can do. It includes showing them why they shouldn’t always do things even when they are capable of doing them. It includes helping them understand right and wrong in the nitty gritty of everyday life where things aren’t black and white. It includes helping them see their true role within their family, within their group of friends, and within broader society. It includes training them to work effectively with others.

So what am I supposed to do? Does my role end when my students and I walk out the door of the classroom? Not a chance. In some ways, that is just the beginning. I can think of two primary reasons – among hundreds – why teachers must participate in school activities with their students.

  1. In those other activities, you are still teaching. A lot. You are still explaining and training and helping students to learn. In fact, you are helping them to learn a whole host of skills and knowledge that cannot be taught in a classroom setting. I think back to my days as a touch football coach, and the many mornings I spent out on the oval teaching my team how to be a team. I’m confident that the lessons I taught out on the field will be some of the most enduring memories of school for the boys and girls I had on my squad. So if you’re a teacher, you should be out there doing that. Why would you pass up that golden opportunity?
  2. In those other activities, you see students in an entirely different light that will help you to understand them better and be a better teacher to them when you are back inside the classroom. Conversely, your students will see you in an entirely different light and that will help them learn more effectively too. Whether it’s sport or photography or army cadets or prefects, each activity you’re involved in helps you see a different side tip your students. Often a student who seems to do badly in your class has meant wonderful skills in other areas, and discovering those will help you appreciate them and give you opportunities to reach out to them (rather than them constantly disliking you because you are merely the teacher of the class they hate the most). Just like air is the medium for sound, I believe that relationship is the medium for learning. And every avenue along which you can form a better relationship with your students will help you to teach them more effectively.

So you can see I have a pretty firm stance on this one. I think it’s a no-brainer to get involved with school activities wherever you can. In fact, hopefully my two reasons above help you to see why I don’t think it’s actually accurate to call them “extra-curricular” activities (since “extra” literally means “outside of” and “separate to”). I think the name “co-curricular” activities is far more appropriate (because it helps us to see that the learning taking place in such activities runs in parallel to the learning taking place in classrooms).

That brings us to the question, then, of time. You wisely ask if these activities are “something a first year out teacher should do, considering the workload”. To be sure, any and all activities you engage in will take a significant investment of time (if you do them well, that is – and you should). There are only so many hours in the day and that means the time you spend on activities will inevitably take away from your time spent on other tasks related to classroom teaching.

But this fact remains true throughout your teaching career; it’s not like you hit some magical point in your fifth or tenth or twentieth year where you say, “Great, I’ve perfectly mastered my classroom teaching now, so with I’m going to devote some of my wonderful newfound spare time to some activities!” That day never comes. Teaching by its very nature is all-consuming. It’s a bottomless pit that will swallow up however much time you throw at it. You can always plan a more innovative lesson. There is always a new tool or technique or technology to try out. You can always spend more time helping out students after class. And there will always be new opportunities to broaden and deepen your understanding of your subject area.

None of these are bad things – they are the reasons why we as teachers never need to get bored! But they do mean, in my opinion, that you shouldn’t wait indefinitely to get involved in school activities. I would wait an absolute maximum of twelve months, because admittedly the very first year is not just your entry into teaching but into full-time work itself, and that can be pretty traumatic. But after a year, it’s time to take the plunge. Yes, it will take some time out of your 24 hours. It’s so worth it, though, that I don’t think you’ll look back.

Awards aren’t evil

A few days ago, @corisel posted this thoughtful reflection on awards in teaching. It’s well written and touches on a really important point: that teaching is a team sport. A single child is raised by a community of educators, not by any one teacher in isolation. So, individualistic awards can be a dangerous thing; they can wrongly emphasise the recognition of one person over the whole group that is really responsible for any positive thing that has been achieved.

The whole discussion reminded me of someone else who was very opposed to the whole system of awards that seems to exist in every field under the sun. That person was the renowned physicist Richard Feynman.

Feynman

Feynman was an incredible scientist, but he was also an amazing personality. If you’ve got 10 minutes and want to marvel at his life, watch this very entertaining video about him by Scishow. The relevant fact about him, though, was that he hated the idea of awards. As in education, science is always a group endeavour. Every discovery stands on the progress and work of others. So, Feynman argued, it was wrong to recognise individuals with awards. In one TV interview he said:

I don’t like honours… I don’t need anything else. I don’t see that it makes any point that someone… should decide this work is noble enough to receive a prize. I’ve already got the prize: the prize is the pleasure of finding the thing out, the kick in the discovery, the observation that other people use. Those are the real things; the honours are unreal to me. I don’t believe in honours. It bothers me. Honours bother me!

His words seem to capture the essence of what many people have been saying so far in the discussion. But there’s more to the story. Feynman was awarded the Nobel Prize in Physics in 1965 – and given his whole stance on things, you would expect him to have refused the prize (he wouldn’t have been the first). However, he didn’t; he accepted the prize. This wasn’t just because he wanted his name up in lights all of a sudden. He actually went through a philosophical reversal about prizes when he saw the response of his friends and the common populace when he was awarded the prize. In his acceptance speech for the Nobel Prize, he said:

And so, you Swedish people, with your honours, and your trumpets, and your king – forgive me. For I understand at last – such things [awards] provide entrance to the heart. Used by a wise and peaceful people they can generate good feeling, even love, among men, even in lands far beyond your own. For that lesson, I thank you.

His point is that awards have a power to draw attention to things that are good, and make people aware and curious about good things that they otherwise would not have done. Yes, they can be abused – they can be pursued for selfish reasons or given out as mere political gestures – but the awards themselves are not bad. In fact, they can be wonderful.

All this is just to think through the other side of things. I have no personal stake in the issue as I’ve never done anything to deserve such awards anyhow! But I thought it interesting and thought it would be nice to hear another point of view (namely Feynman’s – not mine).