Squaring the Circle

I seriously doubt that this estimation method of Squaring the Circle is unique, but it's new to me. For further reading of this topic and many more, I strongly encourage you to purchase the text 

"Squaring the Circle - Geometry in Art and Architecture" by Paul Calter.

download the sketchpad file 

version 5.0 | version 4.07

Unwanted Weed or Saught After Fractal

The mimosa is a very common to Virginia. I've seen this plant my whole life, but today it's as though I've seen it for the first time.

I vote "fractal"!

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Would Fibonacci eat this?

Count the spirals of this cauliflower and decide for yourself!

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Snail Shell

My son Sam found this amazing example of one of nature's spirals. Because of it's circular design, it is best described as an Archimedian spiral.

Thanks Sam!

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Richmond Architecture

Here is a list of 13 samples of Richmond Architecture that demonstrate excellent geometric design. Get in your car and GO!

For more info on my recent VCTM presentation "Architecture is Frozen Music" go here.

Richmond Architecture

The Fan District

Science Museum (Union Station)

2500 West Broad Street, Richmond, VA 23220

All Saints Reform Presbytarian Church

3000 Grove Ave., Richmond VA 23221

Congregation Beth Ahabah

1111 West Franklin Street, Richmond, VA 23220

Cathedral of the Sacred Heart

800 South Cathedral Place, Richmond, VA 23220

Grace and Holy Trinity Church

8 North Laurel Street, Richmond, VA 23220

Richmond Landmark Theater

6 North Laurel Street, Richmond, VA 23220

St. John’s United Church of Christ

503 North Lombardy Street, Richmond, VA 23220

First English Lutheran Church

1603 Monument Ave, Richmond, VA 23220

Downtown

Old City Hall

1001 East Broad Street, Richmond, VA 23219

Virginia State Capitol

920 Bank Street, Richmond VA, 23219

Egyptian Building

1301 East Marshall St, Richmond, VA 23219

Other

St. Joseph’s Villa for Children

8000 Brook Road, Richmond VA, 23227

The University of Richmond

28 Westhampton Way, University of Richmond, VA 23173

Old City Hall

Penrose Tiling

This tiling was created recently by one of my students as part of her tiling project. Excellent Work!!!

For more information on Penrose Tilings: http://en.wikipedia.org/wiki/Penrose_tiling

Leonardo meets Jack - Fibonacci Poetry with a Beat

Leonardo Pisano Bigollo

a.k.a. – Leonardo of Pisa

a.k.a. - Fibonacci

Jean-Louis Kerouac

a.k.a. – Jack Kerouac

a.k.a. – The Icon of the Beat Generation

After reading a recent article from the MAA Focus, a newsmagazine of the Mathematical Association of America on Literature and Mathematics, I was discourage that the mathematics only existed in the subject matter of the poetry. Not to say that the poetry was weak, but the use of mathematics was not impressive. If the building blocks of mathematics are numbers, then why not consider the building blocks of literature the alphabet. One similarity is that both number and alphabet are very fundamental and taught to children are the same time.

  

Another obvious similarity is sequence. The integers are ordered sequentially by a difference of 1 (..., n-1, n, n+1, ...) and the alphabet has been an ordered sequence as early as the 8^th century BC.  Mathematicians were quick to agree on their concept, but it took literary scholars thousands of years to perfect their sequence to our current state of 26 elements[i] (western edition).

The obvious difference is the infinite choice of numbers verses the finite set of the alphabet. Luckily, mathematics devised a method for relating two unequal sets and called it modular arithmetic. Oddly enough, we all use it every day (literally) when telling time. There are 24 hours in the day, but only 12 numbers on the clock. Therefore, the 13^th hour of the day is known as 1 o’clock[ii], 18^th hours of the day equates to 6 o’clock, 23 equals 11, etc. In the form of arithmetic, we have:

10 hours + 7 hours = 17 hours (which converts to 5 o’clock)

or

10 + 7 = 5

This mathematical system is called arithmetic modulo 12.

Applying the same principle to the 26 letter alphabet, where each letter is assigned a number (A=1, B=2, C=3, ...) will derive 27 to equal A, 28 to equal B, on into infinity. With modular arithmetic, the natural numbers[iii] and the alphabet now have a 1 to 1 correspondence. Thus my mathematical palette is now alphabetical! I have chosen to collide these two seemly opposite worlds while under the influence of Jack Kerouac[iv], the infamous beatnik poet and author of “American Haikus”.

Fibonacci Haiku #1: Abacus!

Abacus!

End his mind unhip.

Can’t know null.

Fibonacci Haiku #2: Help could knot.

Angst abound.

Cease. End. Heal my urge.

Help could knot.

A traditional haiku has a pattern of syllables of 5 – 7 – 5. A Fibonacci Haiku (a.k.a. Fib[v]) use a syllable line pattern of 1 – 1 – 2 – 3 – 5 – 8. I have chosen to combine the two into a three-line poem using neighboring Fibonacci sequence numbers. In the above example, I used 3 – 5 – 3. This could expand to 5 – 8 – 5 or 8 – 13 – 8 or even larger. To stay in the style of Jack Kerouac, the smaller format is best.

The other prerequisite for this variation is that each syllable must begin with the letter of the alphabet that corresponds to the Fibonacci sequence. The official sequence begins with the number zero, but the application (there is freedom in art) allows the poet to start anywhere in the sequence. The first poem, Abacus!, uses the sequence 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 (which equates to the letters: A, B, C, E, H, M, U, H, C, K, N) while the second poem’s sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 (which equates to an additional A as a beginning.)

3:       A¹ – b²a – c³us!

5:       E⁵nd h⁸is m¹³ind u²¹n – h³⁴ip.

3:       C⁵⁵an’t k⁸⁹now n¹⁴⁴ull.

3:       A¹ngst a¹-b²ound.

5:       C³ease. E⁵nd. H⁸eal m¹³y u²¹rge.

3:       H³⁴elp c⁵⁵ould k⁸⁹not.

The possibilities are endless, as art mediums should be. Yes, I consider poetry, when read aloud, to be performance art. Especially when accompanied by music and read with passion. I hope to continue down this road and share Fibonacci influenced beatnik poetry. Please share your own ideas or poems on this blog.

Jack Kerouac reading “American Haikus”:



Jack Kerouac on the Steve Allen Show reading his own work “On the Road”:


Introduction of a 1 to 1 correspondence of the natural number to the alphabet by the Jackson 5:

---

[i] The reason for the order of the alphabet is still uncertain to this scholar. Please feel free to share any incite.

[ii] The military operates on its own rules, and I fully support their efforts.

[iii] For this exploration, I will be avoiding zero and negative numbers in order to have a positive experience in my self-created English class.

[iv] Jack Kerouac on the Steve Allen Show reading his own work “On the Road”: http://www.youtube.com/watch?v=QzCF6hgEfto

Jack Kerouac reading “American Haikus”: http://www.youtube.com/watch?v=xJdxJ5llh5A

[v] http://en.wikipedia.org/wiki/Fib_(poetry)

Pelasgian Slate III by Richard Carlyon

This piece by Richard Carlyon is entitled "Pelasgian Slate III" and can be found at the Virginia Museum of Fine Art. My first impression ... there must be more to these rectangles ... why those dimensions? The vertices of the rectangles caught my eye as well. As a group, I see a curve. But which curve? Parabola perhaps.

 

So I took my picture, rotated the image, imported into GeoGebra, and started to play.

Using line GH as the directrix and J (midpoint of CD) as the focus, I have this parabola. It's no surprise that the parabola contains C and D, but it was a pleasant surprise to see it intersect A and F. What about B and E. SO CLOSE! Of course, my picture is not "square" so this is just experimentation. 

To me, it's another way to appreciate art. Below is the quote from the artist that was hung next to the artwork.

"My great-uncle Alfred was forty years old on the day that I was born. He died on my twenty-fourth birthday. During the years I spent with him, I never - ever - heard uncle Alfred speak as much as a single word to anyone. His only vocalizations consisted of beautifully hummed sounds. I learned a lot from uncle Alfred."

Could this be a clue to look for musical ratios? More on that another time.

Patterns in Chaos - Jackson Pollock

This painting is entitled "Number 15" by Jackson Pollock and can be seen at the Virginia Museum of Fine Art. Pollok's intent was certainly not pattern, but is there pattern nonetheless? You decide. Are there zones within the work? (See lesson 4-2 on my Art page.)

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Fruit of the Day

What number is represented by this orange? ... 10!
Count the 'cells' and see for yourself.

Better Question:
Is this true for all oranges?
Is this true for all citrus fruits?
When will someone create a word that rhymes with 'orange'?

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Project: Scale Model of the Solar System

1. Anticipatory Set: How big is BIG? - Watch the first two videos from the group below and post your comments to this blog entry. Are we insignificant? Does it matter? How does this make you feel?

2. Project Directions: Create a Scale Model of the Solar System. - Open the project document. pdf | word doc

3. Extension: Why is Pluto not a Planet? - Watch the next three videos from the group below on "Why Pluto is no longer a planet". Again post your thoughts on this blog.

Mathematical Fiction

There's more out there than "Flatland" and much of it is free. Have a look at this web site.

Waiter! Waiter! There are snails on my plate.

Thanks to the pesto, my wife says that they were delicious. I wouldn't touch them, but I did take a good look along with this picture. Nature's spirals are breathtaking. Nature's flavors are an aquired taste.

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