Mathematics and ...

Squaring the Circle

2011-12-31T14:51:00.005-05:00

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

2011-06-18T08:39:00.001-04:00

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?

2011-05-21T18:41:00.002-04:00

Count the spirals of this cauliflower and decide for yourself!

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Snail Shell

2011-03-19T10:11:00.001-04:00

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

2011-03-13T21:28:00.002-04:00

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

2011-02-27T17:54:00.002-05:00

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

2011-01-15T11:43:00.004-05:00

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)

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

2011-01-12T21:12:00.001-05:00

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

2011-01-11T15:30:00.001-05:00

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

2011-01-11T09:14:00.003-05:00

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

2011-01-09T14:54:00.006-05:00

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

2011-01-02T16:37:00.000-05:00

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.

2011-01-01T16:51:00.002-05:00

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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Problem Solving Books

2010-12-30T00:00:00.000-05:00

If you are going to teach mathematics, then you must teach problem solving skills.

I have found two excellent texts on problem solving. One old (free!) and one that is relatively new (in it's 3rd edition). I recommend them both.

1. The Canterbury Puzzles by Henry Ernest Dudeney. 1908

Project Gutenburg - several digital formats (not pdf)
freepdfdocs.com - download pdf format

2. Crossing the River with Dogs - Problem Solving Strategies. 1st Edition, 1994, Key Press

my resource page

For inspiration and insight as to why you must teach problem solving skills, read A Mathematician's Lament by Paul Lockhart.

Magic Squares

2010-12-29T11:43:00.000-05:00

This work of art is by Paul Calter:

It didn't strike me as a magic square at first. That's what so interesting about it.

More art work by Paul Calter: Calter's Web Gallery

More info on magic squares: Wikipedia

How to Make a Monochord

2010-12-27T10:03:00.000-05:00

A monochord is a single string device with an adjustable bridge thus allowing two pitches to be heard simultaneously on either side of the bridge. Pythagoras' most important work is his study of the ratio of these string lengths. This study led to the development of traditional harmony and the major scale.

These ratios are referred to as the "musical ratios". Applications of the musical ratios can be found in art and architecture. [more info by Paul Calter]

Doodling in Math Class: Snakes + Graphs

2010-12-26T20:33:00.000-05:00

This speaks for itself. Vi Hart has interesting ideas. See more at: http://vihart.com/

Florence, Italy

2010-12-26T15:26:00.000-05:00

Photos that I took on my trip to Florence, Italy as part of my REB award.
"Architecture is frozen music." - Johann Wolfgang von Goethe
web album

Mathematics and Music

2010-12-26T08:02:00.001-05:00

This has become my "white whale". I have found it to be very difficult to collect material in such a way so that both topics are meaningfully addressed. There are many examples of good mathematics that deal with music in a trivial way and vice-versa and I don't want to add to that sort of academic fluff!

Below is a link to a my current outline for a 3 week course. It is in no way finished as I am still adding and deleting content and ideas. I look forward to any comments and/or suggestions.

Outline: http://geometryonline.pbworks.com/w/page/33060523/Math-and-Music

pediment vs. rectangle

2010-12-26T08:01:00.001-05:00

Does the ratio of the pediment relate to that of the rectangle beneath? I think about this everytime I see one. The picture is from Christ Lutheran Church in Fredricksburg, Va.

The Geometry of Nature, Art, and Architecture

2010-12-26T07:59:00.001-05:00

Course Description

This three week course will give teachers of mathematics an opportunity to study geometric design and apply this knowledge in creative ways. In doing so, the learner will gain a perspective of the world that reveals the secrets of proportion and how the greatest artists and architects utilized them to create masterpieces. Emphasis will be placed on the use of technology to import graphics from digital cameras or the web and analyze and/or reconstruct them using dynamic geometry software.

Visit: http://geometryonline.pbworks.com