"This famous drawing by Leonardo da Vinci appeared in the book De Divina Proportione, which Leonardo illustrated for mathematician Luca Paoli in 1509. Leonardo wrote an extensive section on the proportions of the human body in one of his notebooks."
This is the first and only page of the section "Secrets of the Renaissance Man." Of course you can't see what drawing Pappas is referring to here, so let me provide a link:
https://leonardodavinci.stanford.edu/submissions/clabaugh/history/leonardo.html
Pappas explains:
"In his book, he also made reference to the works of Vitruvius, the Roman architect (circa 30 B.C.) who also dealt with the proportions of the human body."
The title, De Divina Proportione, refers to the "divine proportion," which is also known as the golden ratio or Phi. Both Vitruvius and Leonardo believed that Phi = (1 + sqrt(5))/2 appeared in certain ratios of the human body. This is explained at the following link:
https://www.goldennumber.net/leonardo-da-vinci-golden-ratio-art/
- In the distance from the Da Vinci’s guide line drawn at the hairline to the guide line at the foot, the following are all at golden ratio points:
- the navel, which is most often associated with the golden ratio of the total height and not the height of the hairline
- the guidelines for the pectoral nipples
- the guidelines for the collar bone
- In the distance from the Da Vinci’s guide line drawn at the elbow to the guideline at the fingertips
- the base of the hand is at a golden ratio point.
"Leonardo adds, The length of a man's outspread arms is equal to his height."
This estimation appears in a Square One TV song, "Rule of Thumb" by Kid 'n Play. The rappers are trying to measure the length of the floor. One member knows that his height is about six feet, so he concludes that the length of his outspread arms from fingertip to fingertip is also six feet. There isn't a separate video on YouTube for this song, but it does appear at the start of this YouTube video on Math Talk, a spin off of Square One TV. This was recently posted a few months ago:
How can I connect this back to the science class I taught last year? Well, the actual ratios of the human body isn't part of the curriculum, although the human body itself is.
Under the old California standards, here's how a seventh grade life science class was organized. It began with a little chemistry with an emphasize on the elements required for life (hydrogen, carbon, oxygen, and so on). Then the lessons focus on cell structure, DNA, and genes. Next would be evolution and the history of life on earth. This is usually followed by biodiversity, with lessons first on microbes and fungi, plants, and then animals. Within the animal unit, typically invertebrates are covered first, then the various orders of vertebrates -- fish, amphibians, reptiles, birds, mammals. So humans appear last in this section -- but then this is followed immediately by the human body, also known as anatomy. Since I should have followed the California standards for seventh grade, this meant that the unit on the human body should have appeared at the end of the year.
Lesson 0.7 of Michael Serra's Discovering Geometry is called "Islamic Art." This is in the Second Edition -- in the modern editions, "Islamic Art" is Lesson 0.6. Serra begins:
"Islamic art is rich in geometric forms. Islamic artists were familiar with geometry through the works of Euclid, Pythagoras, and other mathematicians of antiquity, and they used geometric patterns extensively in their art and architecture."
Yesterday, I rearranged the Pappas pages so that her first page on knots would fall on the same day as the knots lesson in Serra. But the result of this change is that today's Pappas page is on the human body in art, while Serra's lesson on Islamic art has nothing to do with the human body:
"Many of [Muhammad's] followers interpreted his words to mean that the representation of humans or animals in art was forbidden. Therefore, instead of using human or animal forms for decorations, Islamic artists used intricate geometric patterns."
As usual, the questions I derive from Serra's text instruct the student to create Islamic-style art. This art is based on tessellations.
There are a few interesting things in this lesson. First, Serra includes a sidebar called "Improving Reasoning Skills -- Bagels I." As it turns out, Bagels is an old 1980's computer game. I never played it on my old computer, but as a young child, I actually had an old toy (Speak & Math) which included a version of Bagels (called "Number Stumper"). Here's a link to a modern version of Bagels:
http://www.dst-corp.com/james/Bagels.html
During the Responsive Classroom training at my old school, the presenter actually suggested Bagels as an opening week activity. In her version of the game, the word "Bagels" was replaced with "Nada," but the words "Pico" and "Fermi" were retained (so she called the game "Pico, Fermi, Nada"). Again, I don't post any version of "Pico, Fermi, Bagels/Nada," but if you want, you can use it in your own classroom instead of the "Islamic Art" lesson.
I do however include Serra's project for this lesson, "Geometry in Sculpture." This isn't directly related to Islamic art, though. Instead, he writes about Umbilic Torus, a sculpture. It was created by Helaman Ferguson and used as a trophy for the Jaime Escalante award -- named, of course, for the world's most famous math teacher.
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