Lada 🏡 Prkic

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Geometry, All Around Us

Geometry is all around us, and we are surrounded by myriads of geometric forms, shapes, and patterns. Every living organism and all non-living things have an element of geometry within. Understanding the natural world requires an understanding of geometry.

Where there is a matter, there is geometry. (Johannes Kepler)

For me, geometry is a fascinating subject. As the subject of learning, geometry requires the use of deductive reasoning, a logical process in which a conclusion is based on the multiple premises that are generally assumed to be true (facts, definitions, rules). Learning geometry also develops visualization skills and spatial sense - an intuitive feel for form in space.

Geometry is essential in architecture and engineering fields and mostly used in civil engineering. Thorough knowledge of descriptive geometry is definitely required for civil engineers and helps engineers to design and construct buildings, bridges, tunnels, dams, or highways. It has been, by the way, one of my favourite subjects during my university years.

I use mathematics and geometry almost my whole life. To write about geometry in a way that does not look like a boring lecture is challenging. It is, therefore, my intention to write this post in a manner of a picture book – geometry in words and pictures, because I prefer visualization wherever possible. I shall try to skip formulas, theorems, rules. There will be only a little math.

This piece is a review of some interesting geometric forms, which are particularly impressive to me. 

PERFECT FORMS

Necessary Introduction

Polygons (many sides in Greek) are closed plane figures with straight sides. Regular polygons are polygons whose all sides are equal in length. A regular polyhedron (pl. polyhedra) is a three-dimensional solid whose all faces are regular polygons.

Platonic Solids

These regular polyhedra are called the Platonic solids or perfect solids, named after the Greek philosopher Plato although he is not the first who described all of these forms. 

The Platonic solids are symmetrical geometric structures, bounded by regular polygons all of the same size and shape. Moreover, all edges of each polygon are the same length and all angles are equal. The same number of faces meets at every vertex (corner or point).

Furthermore, if you draw a straight line between any two points (vertices) in any Platonic solid, this piece of the straight line will be completely contained within the solid, which is the property of a convex polyhedron.

An amazing fact is that can be only five different regular convex polyhedra. These perfect forms are:

  1. Tetrahedron
  2. Octahedron
  3. Hexahedron (cube)
  4. Icosahedron
  5. Dodecahedron
_- PLATONIC SOLIDS ~~.

TETRAHEDRON OCTAHEDRON HEXAHEDRON ICOSAHEDRON DODECAHEDRON

 

. FOUR SIDED EIGHT SIDED SIX SIDED TWENTY SIDED “TWELVE SIDED’
vAHRE ©, -* “AAIR VEARTH ~~ VWATER GAETHER
4 FACES 8 FACES 6 FACES K PARI Ye 12 FACES
4 POINTS 6 POINTS 8 POINTS 12 POINTS 20 POINTS
6 EDGES - * 12 EDGES 12 EDGES .~ 30 EDGES 30 EDGES |
A A A
a] 180°x 8 360°x6  .  18°X20 rer]

720° DEGREES ~~ 1440° DEGREES  2160° DEGREES 3600° DEGREES 6480° DEGREES
Platonic solids (Image source: https://joedubs.com/)

Plato was deeply impressed by these forms and in one of his dialogues Timaeus, he expounded a "theory of everything" based explicitly on these five solids. Plato concluded that they must be the fundamental building blocks—the atoms—of nature and he made a connection between five polyhedra and four essential (classical) elements of the universe; tetrahedron → fire, cube → earth, octahedron → air, icosahedron → water, and the dodecahedron with its twelve pentagons was associated with the heavens and the twelve constellations.

Later, Aristotle, who had been Plato's student, introduced a new element to the system of the four classical elements. He classified aether as the “fifth element” (the quintessence). He postulated that the stars (cosmos itself) must be made of the heavenly substance, thus aether. Consequently, ether was assigned to the remaining solid --dodecahedron.

Why Only Five?

Geometric argument and deductive reasoning

Postulates:

  1.  At least three faces must meet at each vertex to form a polyhedron.
  2. The sum of internal angles of polygons that meet at each vertex must be less than 360 degrees (at 360° they form the plane, i.e. the shape flattens out).
4
180° o
3 triangles (tetra) 80° (3x60°) tetrahedron
4 triangles 8 240° (4x60°) octahedron
(octa)
) 20 i
S triangles 300° (5x60°) icosahedron
(icosa)
3 squares 8 270° (3x90°) hexahedron
(hexa)
12
324° (3x108°
3 pentagons (dodeca) (3x ) dodecahedron &

If solid's faces that meet at each vertex are regular triangles, squares, and pentagons, the sum of angles at each corner is less than 360°. Forming a regular solid of hexagons won’t work because hexagon has internal angles of 120°, and in the case where a minimum of 3 hexagonal faces meet at one vertex it gives → 3×120°=360°, thus the shape flattens out.

Consequently, there is no platonic solid formed from hexagons or any regular polygon of more than 5 sides.

The table below is a result based on previous arguments and reasoning. Solids are made only of regular triangles, squares, and pentagons. There are only five possibilities, thus five regular solids. Any other combination is not possible!

G00

The mathematical proof given by Euler's formula confirmed that there are exactly five Platonic solids.

THE BEAUTY OF PLATONIC SOLIDS' GEOMETRY

Spheres and Platonic Solids

Each solid will fit perfectly inside of a sphere → circumsphere and all the angular points (vertices) are touching the edges of the sphere with no overlaps. Nonetheless, the inscribed sphere → insphere touches all the faces.

Icosahedron frame with Cube frame with Octahedron frame with
Icosahedron dual inside Octahedron dual inside Cube dual inside

 

Icosahedron frame with Dodecahedron frame with
Dodecahedron dual inside Icosahedron dual inside
 Inspheres of the Platonic solids (Image source:  https://mathworld.wolfram.com/)

Nested Platonic Solids

Platonic solids have the ability to nest one within the other. The corners of the inner Platonic solid touch the vertices or the edges of the outer solid. The amazing animation below shows the configuration of all five Platonic solids, each fits perfectly inside the other.

 

The video shows how (transparent) dodecahedron opens to reveal a cube inside, which opens to allow a tetrahedron to come out, then octahedron, which opens to reveal the inner icosahedron. All the Platonic solids are harmoniously nested one inside the other.

Duals of Platonic Solids

Each Platonic solid has a dual Platonic solid. If a midpoint (centre) of each face in the platonic solid is joined to the midpoint of each adjacent face, another platonic solid is created within the first.

It occurs in pairs between the solids when the number of faces in one solid = the number of vertices in another.

  • The tetrahedron is self-dual (its dual is another tetrahedron), the only one with 4 faces and 4 points
  • The cube and the octahedron form a dual pair (an octahedron can be formed from the cube, and vice versa), 8 faces in cube=8 points in the octahedron, or 6 points in cube = 6 faces in an octahedron
  • The dodecahedron and the icosahedron form a dual pair (a dodecahedron can be formed from an icosahedron, and vice versa), 12 faces in dodecahedron = 12 points in an icosahedron, or 20 points in dodecahedron=20 faces in an icosahedron
Golden rectangle

 

 

1
¢=1,618

The image above clearly shows how the octahedron occurs from the cube - putting a vertex at the midpoint of each face gives the vertices of dual polyhedron – octahedron. In vice versa, by connecting all midpoints of an octahedron’s faces occurs a cube, like in the image below.

a b

 

a+b

a+bhistoaasaistoh
Platonic solids duals (Image source: http://makerhome.blogspot.hr

The Golden Ratio in Icosahedron and Dodecahedron

The icosahedron and his dual pair the dodecahedron are uniquely connected with the golden ratio by virtue of three mutually perpendicular golden rectangles which fit into both. These mutually bisecting golden rectangles can be drawn connecting their vertices and mid-points respectively.

Tetrahedral Molecular Geometry Octahedral Molecular Geometry

 

Methane Sulfur Hexafluoride

 

—

 

109.5°

A golden rectangle is a rectangle which side lengths are in the golden ratio, 1: φ (the Greek letter Phi), where φ is approximately 1.618.

aqlrR.jpeg

Since the ancient days, geometricians have known that there is a special, aesthetically pleasing, rectangle with width 1, length x, and the following property: dividing the original rectangle into a square and new rectangle, as illustrated in the image above, arises a new rectangle which also has sides in the ratio of the original rectangle.

This curious mathematical relationship, widely known as the golden ratio, was first recorded and defined in written form around 300 BC by Euclid, often referred to as the father of geometry, in his major work Elements

The golden ratio refers to a specific ratio between two numbers which is the same as the ratio of the sum of those numbers to the larger of the two original quantities. The value of the golden ratio is an irrational number, which is 1.6180339887......  (assuming that a is greater than b, and b is greater than zero).

LKXEC.jpeg

 

Fractal Structure of Platonic Solids

Each Platonic solid has its own fractal structure, the same repetitive patterns that fit within each other. The two famous Platonic fractals are the Menger Sponge for the cube and the Sierpinski tetrahedron for the tetrahedron, a three-dimensional analogue of the Sierpinski triangle, also called the Sierpinski sponge or tetrix.

8tuoY.jpeg

The illustrations show Menger sponge after the fourth iteration of the construction process, and a Sierpinski square-based pyramid (tetrahedron) and its 'inverse' after the third iteration. On every face, there is a Sierpinski triangle and infinitely many contained within. 

Of all other Platonic fractals, particularly interesting is the Sierpinski dodecahedron. The image below shows a dodecahedral-cornered supercluster, in which a smaller dodecahedron is placed in each corner of the original solid to get a dodecahedral cluster. The process can be repeated indefinitely.

lce1t.jpeg
Sierpinski dodecahedron, 3rd iteration (Image by David Rosser)

PLATONIC SOLIDS IN NATURE

These regular structures are commonly found in nature, but they are generally hidden from our perception. The first manifestation of Platonic solids in nature is in the shape of viruses. Many viruses have a viral capsid, a protein shell that protects and encloses the viral genome, in the shape of an icosahedron. A regular icosahedron is an optimum way of forming a closed shell from identical subunits.

LzTD7.jpeg
Icosahedral viral capsid and HIV  

Of all Platonic solids only the tetrahedron, cube, and octahedron occur naturally in crystal structures. The regular icosahedron and dodecahedron are not amongst the crystal habit. 

Iron pyrite, known as fool's gold, often form cubic crystals and frequently octahedral forms. Calcium fluoride also crystallizes in cubic habit, although octahedral and more complex forms are not uncommon. 

Tetrahedrite gets its name for its common crystal form - tetrahedron. Sphalerite also occurs in a tetrahedral form.

6IAHn.jpeg
Iron pyrite - cube and octahedron structure (Image by Joel Arem)
5f6IP.jpeg
Sphalerite and Tetrahedrite

All Platonic solids occur in a tiny organism known as Radiolaria, which are protozoa – single-celled organisms widely distributed throughout the oceans whose mineral skeletons are shaped like various regular solids.

K2fxu.jpeg
Image source: www. mathnature.com

These Platonic forms also emerge in the mitosis of a developing zygote, the first cell of the human body.  The first four cells occur by dividing, form a tetrahedron.

Three-dimensional molecular shapes (molecular geometry)

Molecular geometry is the three-dimensional arrangement of atoms within a molecule. Molecules are held together by pairs of electrons shared between atoms known as „bonding pairs“.

A molecule of methane (CH4) is structured with 4 hydrogen atoms (H) at the vertices of a regular tetrahedron bonded to one carbon (C) atom at the centroid. When the central atom has 4 bonding pairs, geometry is tetrahedral. All the angles between the two bonds are about 109,5°.

In a molecule of sulfur hexafluoride (SF6) six fluorine atoms (F) are symmetrically arranged around a central sulfur atom and joining together with 6 bonding pairs of electrons and defining the vertices of a regular octahedron. All the bond angles are 90°.

ah3lV.jpeg

Clustering of the galaxies

Scientific observations made by astrophysics (E. Battaner and E. Florido) have shown that the Platonic solids can also be found in the clustering of galaxies. The distribution of super-clusters presents such a remarkable ordered pattern, like in these octahedron clustering of galaxies in the image below where the identification of real octahedra is so clear and well defined.

AXkEE.jpeg
The two large octahedra closer to the Milky Way (Battaner and Florido, 1997)

There are many more amazing examples showing the occurrence of the Platonic solids in nature.

Above the entrance of the famous Plato's Academy has been engraved a quote of which accurate translation there are still many disputes: "Let no one who cannot think geometrically enter."

On the contrary, I invite you to enter into the world of geometry and comment! Have you ever thought about geometry around us? Do you have favourite forms, shapes or patterns? Do you use geometry in your work?

Sources:

1. Minerals by Name

2. Wolfram MathWorld

3. https://towardsabetterworld.com

4. Alt.fractals: A Visual Guide to Fractal Geometry and Design

5. The Nature of Mathematics

6. The Royal Society

**********
 

If you like this buzz about geometry, please give it a "relevant" or a comment. If you really like it, please share.
 

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Comments
Lada 🏡 Prkic

Lada 🏡 Prkic

11 months ago #73

#96
Thank you, Debasish!

Debasish Majumder

Debasish Majumder

11 months ago #72

absolutely fascinating buzz Lada \ud83c\udfe1 Prkic! enjoyed read and shared. thank you for the buzz.

Lada 🏡 Prkic

Lada 🏡 Prkic

11 months ago #71

#93
Yes it is, Zacharias. Am also amazed by how long this post has remained appealing to readers on both platforms beBee and LinkedIn. It is my favourite post.

Lada 🏡 Prkic

Lada 🏡 Prkic

11 months ago #70

#92
Thank you, Bill!

Zacharias 🐝 Voulgaris

Zacharias 🐝 Voulgaris

11 months ago #69

It's truly inspiring how a post on this topic remains relevant even if it's been four years since it was originally published!

Bill Stankiewicz, 🐝 Brand Ambassador

Wonderful post here thank you for sharing, have reshared with all my Georgia students. REGARDS, Bill Stankiewicz 🐝

Lada 🏡 Prkic

Lada 🏡 Prkic

1 year ago #67

Thanks for re-sharing Mohammed Abdul Jawad

Javier 🐝 CR

Javier 🐝 CR

1 year ago #66

thanks Mohammed Abdul Jawad

Lada 🏡 Prkic

Lada 🏡 Prkic

1 year ago #65

Thank you, Zacharias. The topic is my favourite. As for the images, everything is OK on the desktop, but some images fail to upload on mobile. It is because of the technical glitch caused by changes to beBee's programming. To fix it, I have to resize images. It requires a lot of time going through my other posts where I also noticed the same glitch.

Zacharias 🐝 Voulgaris

Zacharias 🐝 Voulgaris

1 year ago #64

Geometry is closely related to Data Science too, particularly in cluster analysis and various heuristics that explore the relationships of the classes present (in classification problems) in relation to the features at hand. Thanks for the insightful article, btw! I noticed that some images don't show for some reason. Is this a technical glitch or did the images fail to upload?

Frans van Wamel

Frans van Wamel

3 years ago #63

A Great Article about geometry! Platonic form are perhaps the least understood building blocks of out reality @Lada 🏡 Prkic. The seminary in form, shape, spin, space and movement of the Universe continues to amaze me. Platonic Forms happens to have become my favorite topic also of late. Interestingly I came across the following video a little while ago and if you are not amazed by what Franck Chester has discovered in addition to what you alluded us of from an artistic perspective, I am convinced that you will see a yet different addition to these forms from integrating these five platonic forms through Secret Geometry which is archy-typical for each one of us. I have found form the most interesting subject of understanding how nature works. https://www.youtube.com/watch?v=Cd0EptTboHg

Frans van Wamel

Frans van Wamel

3 years ago #62

A Great Article about geometry! Platonic form are perhaps the least understood building blocks of out reality @Lada 🏡 Prkic. The inverted symmetry of form, shape, spin, space and movement of the Universe continues to amaze me. Platonic Forms happens to have become my favorite topic also of late. Interestingly I came across the following video a few weeks ago and if you are not amazed by what Franck Chester has discovered in addition to what you alluded us to from an artistic perspective. I am convinced that you will see a yet different form from integrating the these five platonic forms of Secret Geometry is archy-typical for each one of us. I have found form the most interesting subject of understanding how nature works. https://www.youtube.com/watch?v=Cd0EptTboHg

Frans van Wamel

Frans van Wamel

3 years ago #61

A Great Article about geometry! Platonic form are perhaps the least understood building blocks of out reality @Lada 🏡 Prkic. The seminary in form, shape, spin, space and movement of the Universe continues to amaze me. Platonic Forms happens to have become my favorite topic also of late. Interestingly I came across the following video a few waaks ago and if you are not amazed by what Franck Chester has discovered in addition to what you allude us of from an artistic perspective, I am convinced that you will see a yet different form from integrating the these five platonic forms of Secret Geometry is archy-typical for each one of us. I have found form the most interesting subject of understanding how nature works. https://www.youtube.com/watch?v=Cd0EptTboHg

Frans van Wamel

Frans van Wamel

3 years ago #60

A Great Article about geometry! Platonic form are perhaps the least understood building blocks of out reality @Lada 🏡 Prkic. The seminary in form, shape, spin, space and movement of the Universe continues to amaze me. Platonic Forms happens to have become my favorite topic also of late. Interestingly I came across the following video a few waaks ago and if you are not amazed by what Franck Chester has discovered in addition to what you allude us of from an artistic perspective, I am convinced that you will see a yet different form from integrating the these five platonic forms of Secret Geometry is archy-typical for each of us. https://www.youtube.com/watch?v=Cd0EptTboHg

Frans van Wamel

Frans van Wamel

3 years ago #59

Gerat article. Platonic form are perhaps the least understood building blocks of out reality @Lada 🏡 Prkic. The seminary in form, shape, spin, space and movement of the Universe continues to amaze me. Platonic Forms happens to have become my favorite topic also of late. Interestingly I came across the following video a few waaks ago and if you are not amazed by what Franck Chester has discovered in addition to what you allude us of from an artistic perspective, I am convinced that you will see a yet different form form from integrating the these five platonic forms of Secret Geometry. https://www.youtube.com/watch?v=Cd0EptTboHg

Lada 🏡 Prkic

Lada 🏡 Prkic

4 years ago #58

#76
Thank you so much, Joanne Gardocki. A comment like this one is the best reason for all the effort and time put into writing the post. :-)

Paul Walters

Paul Walters

4 years ago #57

Lada \ud83c\udfe1 Prkic You know, I live with a really clever woman, a really, really clever woman. Was just thinking after I read your piece, ( had to read it twice ) if you were in our immediate circle of friends it would be slightly intimidating...although I could make the tea while you two chat !

Lada 🏡 Prkic

Lada 🏡 Prkic

4 years ago #56

#73
Thank you dear Ali Anani for commenting again. That means a lot, especially coming from such a great thinker.

Ali 🐝 Anani, Brand Ambassador @beBee

ALthough I read this buzz before reading it again sounded as if I am reading it for the first time. You are amazing Lada \ud83c\udfe1 Prkic with your explanations and the way you move your thoughts. As a chemist, I had to do a lot with polygon such as polyhedrons and octahedrons and then on fractals. As much as I knew about them, still your buzz added many new perspectives. Thank you for sharing this lovely buzz.

Lada 🏡 Prkic

Lada 🏡 Prkic

4 years ago #54

#70
Ginger A Christmas, I am amazed at your comment. You can really think geometrically. 🙂 Many students always ask themselves the same question that you asked, “When am I going to use this in my life?” But you had a very wise teacher whose answer I really liked, “Think further, Miss, think further.“ I will remember this. Thank you for sharing this post on other social media. I've started blogging just recently and don't write as much as I would like. This is the post I like the most.

Lada 🏡 Prkic

Lada 🏡 Prkic

4 years ago #53

#68
Thank you Ben. It is one of the best compliments I received on my article. :)

Milos Djukic

Milos Djukic

4 years ago #52

#65
My pleasure Lada Prkic :) Thanks!

Lada 🏡 Prkic

Lada 🏡 Prkic

4 years ago #51

Milos, thanks for making others, who may have missed this post, aware of the beauty of geometry.

Lada 🏡 Prkic

Lada 🏡 Prkic

4 years ago #50

#63
Glad you like it Margaret. The new post about "human forms" is already taking shape in my mind.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #49

#60
I shall try. Am really intrigued by this idea.

Ali 🐝 Anani, Brand Ambassador @beBee

#59
The greatest love is towards the greatest reality- people. This is why I attempt to use science to explain human-related activities. Please find the time to write your promising buzz Lada Prkic

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #47

#57
Thank you very much! You beautifully compiled other people's thoughts, but your idea is a seed for all these thoughts. :) I'd love to have the time to write a post about shapes and colours of humans. I am just writing another buzz related to geometry, which is obviously my first love.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #46

Thank you very much! I'd love to have the time to write a post about shapes and colours of humans. I am just writing another buzz related to geometry, which is obviously my first love.

Ali 🐝 Anani, Brand Ambassador @beBee

#55
Beautiful your comment is dear Lada Prkic as well as your sharing on LI. You add new colors of thinking to the presentation. This presentation is in reality a compilation of beautiful minds.

Ali 🐝 Anani, Brand Ambassador @beBee

Beautiful your commment is dear lada Lada Prkic as well as your sharing on LI. You add new colors of thinking to the presentation. This presentation is in reality a compilation of beautiful minds.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #43

#40
It took me a while to comment on your SlideShare presentation dear Ali Anani, http://www.slideshare.net/hudali15/thinking-shapes-and-colors I tried to contribute in a valuable way. Hope you find it worthy. :-)

Franci 🐝Eugenia Hoffman, beBee Brand Ambassador

#50
Yes, you have and very much so.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #41

#51
I forgot to thank you for your words of praise, Milos. They make me happy. :-)

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #40

#51
I can't thank you enough for promoting my post here and on Pulse, my dear friend. You have shown me a real meaning of social media engagement.

Milos Djukic

Milos Djukic

5 years ago #39

#50
Yes you did Lada Prkic :) Congrats! A masterpiece in science communication.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #38

#49
Many thanks Franci Eugenia Hoffman. I really tried to make it interesting and informative. It seems I have succeeded.

Franci 🐝Eugenia Hoffman, beBee Brand Ambassador

This is an outstanding post Lada Prkic. Well written, comprehensive, and nicely structured. Well worth the read.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #36

Phil Friedman, may I tag you to read this buzz? Tagging people is not my usual habit.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #35

#45
The sacred geometry is an extremely fascinating concept. I agree, Savvy, that it takes time to fully comprehend its principles and applications. Thanks for the link about the celestial DNA time spiral. I am just at the beginning of my fractal journey, but I shall try to learn more about this.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #34

#40
Like you said, dear Ali Anani, some posts that move us require repeated reading, and your presentation is one of them. It takes time to absorb all the thoughts based on you great analogy. I’ve just started to read and I’ll comment as soon as possible.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #33

#38
It would be my pleasure if you re-visit my post, like our friend Ali Anani did. Thank you for the share, dear Savvy Raj

Ali 🐝 Anani, Brand Ambassador @beBee

I have a habit of re-visiting posts that move me. I am visiting this one again. The comments are superb. I just remembered I co-authored a presentation on shapes. I started the initial seed, and then invited other authors to contribute. I think you shall enjoy this one Lada Prkic. http://www.slideshare.net/hudali15/thinking-shapes-and-colors

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #31

#34
Dear CityVP Manjit, your words express exactly what I think about the educational system. Mathematics is not boring and geometry is not incomprehensible, just the opposite, but the teaching methods make them uninterested for the pupils. To understand geometry or mathematics is much more than learning by rote, which is usually the case. It requires them to connect to real life and they should be taught through the examples in nature. Thank you for enriching this conversation with your comment, as always, and thanks for your kind words about the buzz.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #30

#17
I’m glad to meet another female engineer on beBee. Thanks for your kind words, Dilma. I was hoping people would be drawn to this topic if explained in a visual way because geometry itself is a very visual subject. I was also hoping there will be some kind of conversation on this topic. It seems to me that most people think about geometry as a difficult subject, but instead we use geometry in everyday life although are unaware of that fact - like cutting the food for example. :-)

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #29

#33
Totally understand you, Dale. Next time will be less links, I promise. 🙂 I am glad you have read even linked articles. This is what I hoped when I wrote this buzz. You are kind of a reader every author can only wish. Thank you for that.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #28

#17
I’m glad to meet another female engineer on beBee. Thanks for the kind words Dilma. I was hoping people would be drawn to this topic if explained in a visual way, because geometry itself is a very visual subject. I was also hoping some kind of conversation on this topic, but it seems to me that most people think about geometry as a difficult subject, but instead we use geometry in everyday life although are unaware of that fact. - like cutting the food for example. :-)

CityVP Manjit

CityVP Manjit

5 years ago #27

This is a fine example of why mathematics is different from numeracy. The geometrical observations here demonstrates the power and beauty of mathematics. Mathematics is not boring, what bores students is an education system that has turned it into a system of calculation, rather than a way of describing the very constructs of nature. This is what separates statistical thinking from mathematical thinking, because one provides the calculations to create things, the other (mathematics) is a way of seeing, and as such is a language that most of us did not learn through a rote educational system. This failure of educators to show that mathematics is a supreme and beautiful form of education is a shame because the love that Lada Prkic has shown in putting together this superb buzz, is what we most of missed out on as an education, an education system that did not teach us mathematics, only calculation. Mathematics is a 21st Century language that a traditional education system made us illiterate about. This buzz shows me what the vast majority of people like me missed out on as an education.

Mamen 🐝 Delgado

Mamen 🐝 Delgado

5 years ago #26

#30
😘

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #25

#27
You’re wright, Lisa Gallagher To understand nature around us, like rock formation, we use also geometry. I think that within each one of us lurks a small geometrician. :-) Thank you for your kind words of praise.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #24

#26
I'm so glad you like it, Mamen Delgado. By the way, dear Madam Ambassador, I like your new picture. :-) Thanks for your support.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #23

#25
#24 Thank you for the share and comment my dear Madam Ambassador. :-)

Lisa Gallagher

Lisa Gallagher

5 years ago #22

The video was quite capturing! I love to look at rock formations, especially those near water. I have many photos I've taken with different rock formations. Some of my photos I take in the winter so i can capture ice formations frozen on the sides of the rocky cliffs. So, in this sense maybe thats where geometry may play a role for me, as in fractals? Excellent buzz Lada Prkic, your extremely intelligent!

Mamen 🐝 Delgado

Mamen 🐝 Delgado

5 years ago #21

Wooow Lada Prkic... What an impressive buzz!!! It's an intense and powerful lesson about Geometry!! 🔯 🔯

Lisa Gallagher

Lisa Gallagher

5 years ago #20

Sharing Lada Prkic thought provoking buzz. Read the comments too, insightful!

Lisa Gallagher

Lisa Gallagher

5 years ago #19

Oops hit send before I was done, I will read this though, I might learn a thing or two!

Lisa Gallagher

Lisa Gallagher

5 years ago #18

#22
I'm going to be honest Lada Prkic, math was never my strong point. I dont remember Geometry.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #17

Lisa Gallagher, are you interested in a bit of geometry?

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #16

#20
Exchanging thoughts with a thinker like you, Ali Anani, also send my mind into new territories. I'm grateful for such an opportunity.

Ali 🐝 Anani, Brand Ambassador @beBee

#18
Lada Prkic- thank you so much for the link and I never expected that imagining the human behavior as a fractal would send my mind into new territories.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #14

#16
Dear Debasish, your comments sound like your beautiful poems. Math, geometry, and poetry have much in common. Inspiration and imagination are needed in poetry as well as in geometry and mathematics. Many poets have also been mathematicians. "Pure mathematics is, in its way, the poetry of logical ideas." So, you are a mathematician, in a way, but you are not aware of that. :-)

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #13

#15
Many thanks for your comment Ali Anani. I’m so glad that you like the visuals, which are an essential part of this buzz. As I said at the beginning of the post, I prefer visualization wherever possible because I am pretty much a visual learner. While I was writing this buzz I was also doing some research on post topic and came across an interesting article dealing with a statement that human behaviour follow the same fractal patterns which are the building blocks—atoms--of the natural world. This applies not only to the physical world, but also to the world of human emotion and behaviour. Hence human behaviour is fractal in nature. An eye-opening article worth reading, https://onefootwalking.wordpress.com/2011/01/29/fractals-and-human-behavior/ Yes, we are, indeed, fractals of behaviours.

Ali 🐝 Anani, Brand Ambassador @beBee

This is a beautiful buzz Lada Prkic. We are all molecules of different shapes to do certain functions. Molecules have shapes. Minerals have shapes. I wonder if humans behavior have also their molecular shapes. This is a beautiful buzz with the visuals enriching it tremendously. The examples are well-chosen. The explanation of fractal shapes is excellent. The embedded video is excellent. I feel like saying we are fractals of behaviors- behavior within behavior within behavior

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #11

#12
Praveen, I think your comment about bubbles inside bubbles is just the right cue for a comment of our friend Ali Anani.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #10

#12
Praven, I think your comment about bubbles inside bubbles is just the right cue for a comment of our friend Ali Ali Anani.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #9

Milos Djukic thanks a lot for the share. I'm glad you like it!

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #8

#9
“Fractality” is only one of the Platonic solids amazing properties. To me, the nesting of the Platonic solids one inside the other is the most intriguing feature. This video simply thrilled me. Thanks for mentioning the effort, dear Praveen Raj Gullepalli. I have to say that I really enjoyed writing this post. I made it without haste and wrote only when I was in a writing mood.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #7

#4
Thanks for the share, Aurorasa.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #6

#3
Dean Owen, I sincerely appreciate your kind words about my buzz. While I was preparing the post I came across a lot of sites on how to build models of the Platonic solids using sheets of paper or other materials. It’s really interesting for the kids like me. :-) By the way, all these solid names seem so mystical and I like how they sound.

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #5

#2
Excellent modification of the famous quote, Ken. Love, like music, has its own geometry. Everything is interconnected. :-)

Lada 🏡 Prkic

Lada 🏡 Prkic

5 years ago #4

#1
Thanks Gert! I put a lot of efforts into this buzz. Sadly, some editing options are not displayed as in the draft.

Dean Owen

Dean Owen

5 years ago #3

Brilliant! I used to love the word dodecahedron. I remember making one out of coloured paper and hanging it in my bedroom. Fascinating article Lada Prkic.

Ken Boddie

Ken Boddie

5 years ago #2

All I can say, Lada, is WOW!!!!!!! "If geometry be the food of love, play on".

Gert Scholtz

Gert Scholtz

5 years ago #1

Lada Prkic for his interest.

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