Bathroom Math

i was in the bathroom appreciating the rosette patterns on the shower curtain and then i realized the tiles are a frieze pattern and the plant is a fractal... math is everywhere!

More Fractals

http://www.youtube.com/watch?v=gXVj2SLSnSc&feature=related

So fractals and all the shapes they are in and can become have really started to amaze me. Here's another video showing how with simple manipulation, fractals can change from one thing to the next.

Huge Eggs?

Speaking of scale, check out these huge eggs!  I thought they were pretty funny.

fibanocci sequence/Human body fractals

I came across this charm with the fibanocci sequence and never would of considered it a fractal until now. It fits its description and relates to more nature and patterns in everyday life, like in plants growth and in our body portions.

Human Body

If you are still not convinced that that fractals, being a math topic, are very important in real life, your opinion might change after finding out that you yourself are made of fractals!

THE LUNGS

The first place where this is found is rather obvious to anyone who knows fractals — in the pulmonary system, which you use to breathe. The pulmonary system is composed of tubes, through which the air passes into microscopic sacks called alveoli. The main tube of the system is trachea, which splits into two smaller tubes that lead to different lungs, called the bronchi. The bronchi are in turn split into smaller tubes, which are even further split. This splitting continues further and further until the smallest tubes, called the bronchioles which lead into the alveoli. This description is similar to that of a typical fractal, especially a fractal canopie, which is formed by splitting lines:

The endpoints of the pulmonary tubes, the alveoli, are extremely close to each other. The property of endpoints being interconnected is another property of fractal canopies.

THE ALVEOLI

Another supporting evidence that your lungs are fractal comes from measurements of the alveolar area, which was found to be 80 m² with light microscopy and 140 m² at higher magnification with electron microscopy. From the geometric method we know that the increase in size with magnification is one of the properties of fractals!

THE BLOOD VESSELS

Similarly to bronchial tubes, splitting can also be found in blood vessels. Arteries, for example start with the aorta, which splits into smaller blood vessels. The smaller ones split as well, and the splitting continues until the capillaries, which, just like alveoli, are extremely close to each other. Because of this, blood vessels can also be described by fractal canopies.

THE BRAIN

The surface of the brain, where the highest level of thinking takes place contain a large number of folds. Because of this, a human, who is the most intellectually advanced animal, has the most folded surface of the brain as well. Geometrically, the increase in folding means the increase in . Instead of 2, which is the dimension of a smooth surface, the surface of a brain has a dimension greater than 2. In humans, it is obviously the highest, being as large as between 2.73 – 2.79. Here’s another topic for science fiction: super-intelligent beings with a fractal brain of dimension up to 3!

MEMBRANES

The surface folding similar to that of a brain was found in many other surfaces, such as the ones inside the cell on mitochondria, which is used for obtaining energy and the endoplasmic reticulum, which is used for transporting materials. The same kind of folding was found in the nasal membrane, which allows sensing smells better by increasing the sensing surface. However, in humans this membrane is less fractal than in other animals, which makes them less sensitive to smells.

The fractal dimension of some anatomical structures are given below. Note that all dimensions are greater than you would expect them to be, and most are fractions, which automatically implies that the structures are fractal.

Anatomical Structure	Fractal Dimension
Bronchial Tubes	very close to 3
Arteries	2.7
Brain	2.73 – 2.79
Alveolar Membrane	2.17
Mitochondrial Membrane (outer)	2.09
Mitochondrial Membrane (inner)	2.53
Endoplasmic Reticulum	1.72

Fractals, in addition to the anatomical structures above can be found in the body on smaller scales in various molecules.

fractal inspired tattoos

Any subject that I google, its almost positive that someone has it tattooed on themselves. Here are two fractal inspired tattoos, for your viewing pleasure.

create your own snow flakes/fractals/wallpaper patterns.

I came across this drawing site and your able to create so many different types of images. check them out and try one yourself.
http://www.myoats.com/create.aspx.

more unique fractal art

I did a general google search for fractal art and I came upon these, which i find very unique to the ones Ive come across so far. Enjoy.

Math Anxiety

I found a site that helps a TON with coping with math anxiety, which is very helpful right before final exams.

TIps for dealing with math anxiety

When I was searching for things to blog about, I found Pentomino, which are shapes each made of five squares attached to each other in varying ways. The blocks are then tiled on a plane to create really interesting and stimulating patterns.



The building blocks


A few basic designs

They really remind me of Tetris, mostly.

Fractal Cow?

Just look at this!

Here you can find the blog dedicated to this cow.

This is what they are saying about it:

The Fractal Cow projects the contradictory vision of contemporary architecture that fractal spaces of nature are being replaced by the artificial fractal spaces of our society.

The conditions of our world's interconnected, hyperactive and dense space can be explored as we perceive it, with its inherent fractal dimensions and qualities exceeding the Cartesian concept of space that has dominated architecture and sculpture.

The underlying structure and labyrinthine networks of the city's architecture are artificial representations of the forms of nature and threaten to obliterate natural environments and vegetation at the same time.

Fractals in nature

I have been looking at all kinds of fractals that occur in nature, and I thought that there were some very interesting finds, including:

Romanesco Broccoli



Trees and plants, specifically the Fern


blood vessels

Wear your Fractal

I found a company that makes clothing and other oddities with fractal designs.  They are actually pretty cool.  Check it out here.  

Exam Fun

I thought everyone would appreciate a little animal humor from Wired Science during the busy week of exams.  Enjoy!  http://blog.wired.com/wiredscience/2008/12/top-10-incredib.html

This is a pretty funny video about the number zero.

This is a really cool fractal of a dragon. It's a really short video, but I enjoyed it.

For those interested in Knitting and Crocheting, (Alyssa) I found this site which explains the correlation between knitting and math, and how some people use math to answer knitting problems, while some use knitting to answer math problems. On the right side of the site there is a mini-slide show that includes a picture of a mobius strip that someone knit.

Egyptian Fractals

This image compares Egyptian architectural designs to the Cantor set, a fractal pattern. The Egyptian design is based on the symbol for biological recursion of the lotus flower. Perhaps this design influenced Cantor, who was fascinated with ancient Egypt.

Art and Fractals

We coexist harmoniously with the fractal patterns in nature, our visual systems process thoughts that themselves have become fractalized, subconsciously in tune with the world around us. In so doing, we have inadvertently added this familiarity to our surroundings by letting fractals permeated our art, architecture, music, and even popular culture.
We can compare a 1920 Mondrain abstract composition with its use of rectangles to a Longone-Birni settlement in Cameroon to the first two stages of a fractal generated piece. This geometric style referred to as Neo- Plasticism, influenced many of architectural and sculptural designs that we now see in the modern world.

Fractal Poem

"To see the world in a grain of sand,
And a heaven in a wild flower,
To hold infinity in the palm of your hand,
And eternity in an hour."

- William Blake¹, Auguries of Innocence, 1803

Because da Vinci was obsessed with finding order in the universe, mathematics was a huge part of his life.  He said, "the point is the first principle of geometry and no other thing can exist in nature or in the human mind from which the point can originate."  He said other principles of geometry are the line, the body, and the body clothed by these surfaces.  

The golden ratio, or golden spiral, has been used in art for centuries to aid in establishing composition.  The shape is also frequently seen in nature and various other places.  Here is an example above.

Art forgeries are often detected using mathematical techniques.  Various equations were created based on time, age, decay, color retention, etc.  This became a popular way to detect forgery with the issue between van Meegeren and Vermeer.

The entire article can be found here.

Angle Calculator

I found this cool angle calculator online that negotiates focal lengths of lenses and film formats.  Click

Math and Painting

I found this interesting article about the relationship between mathematics and painting.  There are a lot of great relationships made.  The article discusses van gogh, kandinsky and fractals.

CLICK

Holiday Fractals

Get 4 large shiny Christmas balls, some coloured wrapping paper, some lights from the Christmas tree, and you have you own fractal laboratory to amuse yourself with over the festive season.

The basic setup is four identical, highly reflective balls sitting is a pyramid formation so that each ball touches every other ball. You can look into the gaps between three balls and either shines light into all or one of the other gaps or places colored paper over the gaps and shines light into the same gap one is looking into (or variations of the above). The resulting of reflections, fractal, has what is known as the Wada property after the Japanese mathematician who studied these spatial divisions in 1917. The Wada property refers to cases when there are three basins of attractions so convoluted that every point on a basin boundary is also on the boundary of all other basins.

More Fun Self-Similar Shapes.

http://www.youtube.com/watch?v=A4MZubCt87s



I found this video looking for more cool patterns created by self-similar shapes. Some of the patterns in here we have discussed like snowflakes, and other we haven't touched upon, like planets, leaves, and shells.

Fractals built Temples

I was researching the Indian temples of Khajuraho when I came across a scholarly article that said fractal geometry was the underlining structure for the building of these temples.  To be honest, I was surprised!  Here is a photo of one of the temples.  

flower patterns

i found this pretty cool program where you can make flower patterns. it reminded me of the wallpaper symmetry groups.

Math in Motion

I found this fun little animation called the chaotic pendulum. It is made with two springs, and the motion of one pendulum keeps the other one moving in a circle. Check it out!

Sumo paint

I stumbled on this website that has an art-making program kind of like paint. The difference is that the brushes are cooler, and it has a symmetry tool (the one that looks like a snowflake). The symmetry tool allows you to create a rosette, kind of like the programs we used in class earlier in the semester. I used the symmetry tool to make some cool pictures. Give it a try!

cactus

I have a grafted cactus on my windowsill and it reminded me of fractals.  Many flowers follow the fractal and Fibonacci sequence patterns.  Sunflowers are also known to be true to this fact.  

This escher fractal drawing is called "Fish and Scales" and was completed in 1959.  Here we see the scales of the large fish developing into fish.  The occurs from both sides.

Angles and Focal Lengths

This chart demonstrates different angles of view from lenses of different focal lengths.  On a camera, lenses of different focal lengths distort and manipulate the image in various different ways depending on how long or wide it is.  

Aperture and Math

Aperture is a space through which light passes in a photographic instrument.  Light gathering power is determined by the area of the lens or mirror.  The area of a circle equals pi times the radius squared.   

Bellows Extension Compensation

When doing close up work in large format photography, there are several mathematically based formulas to figure out how to compensate for the extra space that the light has to travel.  Because the light has to travel further to hit the film plane, typical light meter exposure has to be adjusted. Here are a few formulas:

BE=bellows extension  FL=focal length    

Zero to Infinity in Nothing Flat

On helasculpt.com, you can see the sculpture of Helaman Ferguson, who expresses mathematical equations through sculpture.  His process is equally weighted with the mental (mathematics) and the physical (sculpture).

I found this artist, Eric Hammel, who makes digital art from fractal dimensions.  A writer of military history books, Hammel became interested in this type of art when Kai's power tools plug in was first released for windows photoshop.  Now he uses applications such as Ultra Fractal and Apophysis.  A 8.5 x 11 print sells for $75!

http://www.youtube.com/watch?v=J-fcRzvRBqk&feature=related

Infinity + mirrors = fractals

I just moved into this nice new apartment- already furnished and with several large mirrors hanging. I just realized last night that two big mirrors face each other at either end of the main hallway- you can basically look into infinity in either one although they're only about 15 ft away from each other. The images are self-similar and repeat endlessly...
It reminded me of this famous Dutch painting although i guess Jan Van Eyck's mirror doesn't repeat endlessly- BUT while i was searching for the image i also found this intersting image/info about another mirror.
Tibetan, 17th Century
Divination Mirror of the State Oracle of Tibet
"Seven metal" silver alloy, gold, and turquoise.
"When advising the Dalai Lama, the State Oracle wears an elaborate costume, including this mirror on his chest. In deep trance, he looks into the mirror, not directly, but through another mirror held by an assistant in front of him. Through this doubly distant vision he foretells the future and gives advice."

Lego Sculptures

These mathematical lego sculptures were made using computer code, which reminded me of what we've been learning about fractal computer programs in class. I think these are especially impressive because the shapes are so curvy even though they are made up of lots of small rectangular blocks. Also, legos are awesome.

the social aspect of art

this is to comment on kcantor's comment on the email conversation that David Thorne had about the drawings. this is a piece that speaks to the very soul on the topic of how can one price art. the worth of this piece to begin with is about 27 million or something. it has diamonds all over it. the point trying to be made by the artist is art has an imaginary number on it. the amount of art is never specified to something like the tools used or how much paint was used but rather it just had a random number placed on it, and yes people thought about the time and effort placed onto their piece and gage that for the amount. however with this piece there is automatically a price tag on the piece, without time placed on making it and the effort of the artist. so as you can see art is less of substance anymore its more about the name behind the piece of late. so what is the true problem today. society or one's self as an artist?

foreign fruit

Ever seen a califlower like this before? I typed in fractals in nature and this was one of the first images it presented. It looks almost alien to actually eat this.

Pruning

So i was thinking about fractals and trees. These are an obvious example that we see all the time (especially now that the leaves have fallen the shape of the tiny twigs are so clear and resemble the trunk and main limbs)
so Pruning trees... what does that do to the fractal structure? how do trees compenate for this interference in their natural pattern?
Also Strawberries are amazing and the way they propagate is to send out runners which then take root and grow up to be big enough to send off their own runners. In this way, they are self-similar and continue in an endless fractal-like pattern of growth (more strawberries for everyone!)

Our group could have so made this if we had enough time and bigger sheets of cardboard.

I was wondering why we should learn about fractals so I decided to search online to see if it had any relavcence or daily use. To me the images look a little silly and I didnt understand how the Math could be used. But I found this web site that talked about technology that was made through the knowledge of fractals to monitor fluid movements. I don't really understand all of what they say but check it out.

www.arifractal.com

Web Cams

Hey! I am looking for a web cam to be able to communicate with volunteers in Africa. It has to be suitable for a Mac computer. Does anyone have any suggestions?? I am clueless.

Thanks!

New Wind Turbine Design

A more attractive wind mill design has been made for rural and urban environments. It is pretty cool and definitely eco-friendly. One of these towers can provide 25-30% of the electricity needed to run a house!

If you haven't read this, do yourself a favor and do so:

Below is the complete email conversation that Adelaide man David Thorne claims he had with a utility company chasing payment of an overdue bill.

From: Jane Gilles
Date: Wednesday 8 Oct 2008 12.19pm
To: David Thorne
Subject: Overdue account

Dear David,
Our records indicate that your account is overdue by the amount of $233.95. If you have already made this payment please contact us within the next 7 days to confirm payment has been applied to your account and is no longer outstanding.

Yours sincerely, Jane Gilles

From: David Thorne
Date: Wednesday 8 Oct 2008 12.37pm
To: Jane Gilles
Subject: Re: Overdue account

Dear Jane,
I do not have any money so am sending you this drawing I did of a spider instead. I value the drawing at $233.95 so trust that this settles the matter.

Regards, David.

From: Jane Gilles
Date: Thursday 9 Oct 2008 10.07am
To: David Thorne
Subject: Overdue account

Dear David,
Thankyou for contacting us. Unfortunately we are unable to accept drawings as payment and your account remains in arrears of $233.95. Please contact us within the next 7 days to confirm payment has been applied to your account and is no longer outstanding.

Yours sincerely, Jane Gilles

From: David Thorne
Date: Thursday 9 Oct 2008 10.32am
To: Jane Gilles
Subject: Re: Overdue account

Dear Jane,
Can I have my drawing of a spider back then please.

Regards, David.

From: Jane Gilles
Date: Thursday 9 Oct 2008 11.42am
To: David Thorne
Subject: Re: Re: Overdue account

Dear David,
You emailed the drawing to me. Do you want me to email it back to you?

Yours sincerely, Jane Gilles

From: David Thorne
Date: Thursday 9 Oct 2008 11.56am
To: Jane Gilles
Subject: Re: Re: Re: Overdue account

Dear Jane,

Yes please.

Regards, David.

From: Jane Gilles
Date: Thursday 9 Oct 2008 12.14pm
To: David Thorne
Subject: Re: Re: Re: Re: Overdue account

Attached

From: David Thorne
Date: Friday 10 Oct 2008 09.22am
To: Jane Gilles
Subject: Whose spider is that?

Dear Jane, Are you sure this drawing of a spider is the one I sent you? This spider only has seven legs and I do not feel I would have made such an elementary mistake when I drew it.

Regards, David.

From: Jane Gilles
Date: Friday 10 Oct 2008 11.03am
To: David Thorne
Subject: Re: Whose spider is that?

Dear David, Yes it is the same drawing. I copied and pasted it from the email you sent me on the 8th. David your account is still overdue by the amount of $233.95. Please make this payment as soon as possible.

Yours sincerely, Jane Gilles

From: David Thorne
Date: Friday 10 Oct 2008 11.05am
To: Jane Gilles
Subject: Automated Out of Office Response

Thankyou for contacting me. I am currently away on leave, traveling through time and will be returning last week.

Regards, David.

From: David Thorne
Date: Friday 10 Oct 2008 11.08am
To: Jane Gilles
Subject: Re: Re: Whose spider is that?

Hello, I am back and have read through your emails and accept that despite missing a leg, that drawing of a spider may indeed be the one I sent you. I realise with hindsight that it is possible you rejected the drawing of a spider due to this obvious limb ommission but did not point it out in an effort to avoid hurting my feelings. As such, I am sending you a revised drawing with the correct number of legs as full payment for any amount outstanding. I trust this will bring the matter to a conclusion.

Regards, David.

From: Jane Gilles
Date: Monday 13 Oct 2008 2.51pm
To: David Thorne
Subject: Re: Re: Re: Whose spider is that?

Dear David, As I have stated, we do not accept drawings in lei of money for accounts outstanding. We accept cheque, bank cheque, money order or cash. Please make a payment this week to avoid incurring any additional fees.

Yours sincerely, Jane Gilles

From: David Thorne
Date: Monday 13 Oct 2008 3.17pm
To: Jane Gilles
Subject: Re: Re: Re: Re: Whose spider is that?

I understand and will definately make a payment this week if I remember. As you have not accepted my second drawing as payment, please return the drawing to me as soon as possible. It was silly of me to assume I could provide you with something of completely no value whatsoever, waste your time and then attach such a large amount to it.

Regards, David.

From: Jane Gilles
Date: Tuesday 14 Oct 2008 11.18am
To: David Thorne
Subject: Re: Re: Re: Re: Re: Whose spider is that?

Attached