Category: Numbers and Sequences

Golden Mean Ratio: Egyptian Sculpture

Math in Ancient Sculpture is one of the most interesting and intriguing applications of mathematics, both formula equation and geometric structure.  This article examines the Golden Mean Ratio and Fibonacci FibonacciChamomile wikip Oct 7 2013  Nefertiti Neues Museum Wikip Oct 7 2015Sequence found in the Bust of Nefertiti in Egypt in the early 1900’s. Nefertiti literally, means, “the beautiful one has come”, and is the 14th-century BC Great Royal Wife (chief co-regent) of the Egyptian Pharaoh Akhenaten of the Eighteenth dynasty of Egypt, 1352 BC to 1336 BC.  The iconic Bust of Nefertiti is in Neues Museum of Berlin; though originally in Egypt (En.   Wikipedia.   Org / wiki/Nefertiti_Bust   (230px-Nofretete_Neues_Museum 2 .jpg)  During excavation of Nefertit’s Bust, one of the items found noted Thutmose as being the court sculptor of Egyptian Pharaoh Akhenaten, Nefertiti’s Royal Husband.  In Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science, Stakho speaks of the Berlusian philosopher Edward Soroko’s attempts to determine what mathematical ideas were used in the creation of Her Bust, “…Harmony was the perogative of the Divine order that dominated the universe, and geometry was the main tool of its expression. 

Nefertiti, playing role of Goddess, thus Her image personifying the Wisdom of the world, must have been formed with geometrical perfection and irreproachable harmony, beauty and clarity.  As a matter of fact, the main idea of ancient Egyptian aesthetic philosophy was to glorify the eternal, the measured, and the perfect in a constantly changing universe…In his analysis, he found a harmonious system of regular geometric figures such as triangles, squares, and rhombi.”  (Alekseĭ Petrovich Stakho; Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science; Scott Anthony Olsen. Book. English. 2009; pp 57-59)  Soroko believed that the Bust of Nefertiti was created based upon the Geometries of the Golden Mean Ratio and the Fibonacci Sequence.  A graphical image of Her Bust, as was drawn by Soroko is shown in this link:  Highly recommended to click on this link to have a finer and more detailed idea of the geometries that Thutmose is thought to have in mind when sculpting Her Bust.  One can appreciate, in particular, how the front top vertex of Her headdress is in perfect alignment with Her Heart and Breast Bone.

Golden Mean Squares Wiki Oct 7 2014Depicted here, the Golden Mean Ratio, and in the geometries thought to be applied by Thutmose, the Egyptian Court Sculptor, in the creation of Nefertiti’s Bust:

n is to m, as m is to n − m,

or, algebraically,


n            =         m

_____            ______

m                     n – m

A more clear example of the pattern he believed Her Bust was created upon, the Fibonacci Sequence, can more easily be seen in examining nature’s pattern in pine cones, and seashells.  This ratio is also the most efficient mathematical equation for trees absorbing the most amount of sustenance from the sun’s rays, hence growing in a spiral.  In the same way, today, we can apply solar panels onto rooftops mimicking this pattern to most efficiently absorb the sun’s rays.  For further discussion on Fibonacci Sequence, visit here:

See in the photo at the top, the chamomile spiral, as well as in this link on “fibonacci-sequence” showing the pattern.  Adding two consecutive numbers from the sequence to equal the next one following, the basic mathematical sequence looks like this:

0,         1,            1,            2,            3,            5,            8,            13,            21,            34,            55,            89,            144,            233,            377, etc., etc.

That is:



2+1 (the “number” before)=3

3+2 (the number before)=5








etc., etc., etc.

Whereas the fibonacci sequence, cannot technically be exacted in real time, (similarly, there is no real exact end found as of yet to pi’s “3.14159265359…..”), mathematicians drew swirling squares around the spiral in the attempt to exact the formula – basically giving the left side of the human brain a way of understanding the right-brained infinite spiral, being that the left side of the brain requires finiteness / exactness to feel satisfied, shall we say.  

What kind of intriguing application can your class find to prove that math is interesting and fun?

We wonder if the geometries of Nefertiti’s Bust were also the same geometries of Nefertiti’s real-life human head.  The skulls of Ancient Egyptians, as well as those of Ancient Greece and other cultures, have definitely been shown to have geometrically perfect structures.  How can that be?

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Developing Deductive Reasoning with Hula Hoop

Here is a fun game to help students in your elementary math classroom – develop their observation skills while at the same time practice their deductive reasoning. My students have named this game “Soup,” and we pretend that we are cooking up a delicious soup. Feel free to adapt it to your own students’ interests.

Materials: Attribute blocks(these are our ingredients) and a hula hoop (this is our pot in which to cook).


How to Play the Game:

1)    Have your students sit around the outside of the hula hoop so that they can all see and reach it. The teacher begins the game by creating a rule for the “soup” (e.g. square soup). Without telling the students the rule, the teacher places one attribute block into the center of the hula hoop, saying “This piece belongs in my soup today.”

2)    The first student in the circle chooses any other piece, places in in the “pot” and asks, “Does this belong in your soup today?” If the piece matches the rule, the teacher says, “Yes it does,” and the student gets another turn. If it does not, the student removes that piece from the center, and her turn is over.

3)    Students continue to take turns going around the circle. A student may guess the rule only during her turn. (e.g. “I think you are making blue soup.”) If the student is wrong, her turn is over. If she is correct, she wins the game.

4)    You can continue to play the game by creating a new rule or allowing the winning student to create a new rule for her classmates to figure out.


1)    Attribute blocks are excellent tools for this game because they contain four different attributes (color, shape, size and thickness). When I play with very young students, I choose only one attribute (e.g. red soup or triangle soup). However, when I play with older students, I use several attributes (e.g. thick yellow soup or small red triangle soup).

2)    When allowing students to create the rule and start the game, it is a good idea to have them whisper their “soup recipe” in your ear. They tend to forget their rule and provide false information at times!

3)    I find that continuing around the circle after a game is won keeps students from arguing about whose turn it is and gives everyone a chance to play. For example, if the sixth child in the circle correctly guessed the soup recipe, the next game starts with the seventh child in the circle.

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Flower Geometry on Summer Vacation

Summer can be fun Learning Math on your camping trip or at home in your backyard.  An interesting approach to learning geometry in elementary math is by the study of flowers found in your backyard (or schoolyard in Spring or Fall).  Have your class walk around the local grounds, or give them an assignment to go home and document in their notebooks, different flowers, counting the number of petals in the flowers.

A second item, if time permits is to have the Learners identify the flowers as well.  Upon arrival back in the classroom, Learners identify, with their geometry charts, which geometric pattern or shape the flower has grown into.



Eyes  (peepers for finding flowers)

Camera, if available


Pencil and Good Eraser

Pencil Crayons in various colors


Compass (if you wish to measure angles in the shapes)



Next, Learners draw in their notebooks the geometric shape the flower is equated with, and beside the shape, a simple drawing of the flower, coloring the flower drawing with the corresponding color of the petals.  If possible, 3-D forms can be cut out and interlocked together, with a string glued into the top of the start and made into Christmas ornaments.


Yellow Blue-eyed Grass:  (photo Above) 6-Petaled Yellow Wild Flower (that also grows in Bluish Purple and White); found in tall grasses who / that opens up only with the sun, and closes at the end of the day when the sun sets, or on cloudy days.

Geometric pattern:  6-pointed Star Tetrahedron; Two 3-Dimensional Interlocking Equilateral Triangles with a conjoining dot in the middle.  These 2 photos show the star tetrahedron (6-pointed) both in 2-dimensional form (as it would be if drawn flat on a piece of paper).  The second photo is a rendition of a 3-dimensional form (as if it were hanging as an ornament in a tree).   SourceURL:file:///Users/sheila/Desktop/Summer%20Flower%20Geometry.doc

When looking, aim for the pattern that is found when counting the Number of Petals in the Flowers.  In the Yellow Blue-Eyed Grass, there are 6 petals which if gazed at in a 3-Dimensional way, one can see the pattern of the 6-pointed star tetrahedron.

This can be a fun activity to do while on summer vacation – or during the schoolyear in Spring and Fall, and depending what climate area your school is, it can be done during winter as well.

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The Fibonacci Sequence

Fibonacci is a Number / Integer Sequence, that when applied in geometrical form, manifests in a Spiral as in that of a Pine Cone or a SeaShell.  The sequence was named after an Italian mathematician known as Leonardo of Pisa (or Leonardo de Fibonacci).  In 1202, he wrote a book called Liber Abaci in which he gives name to the number sequence.  There are historical examples of the sequence showing up in East Indian mathematics as well.

Add two consecutive numbers from the sequence to equal the next one following.  The basic sequence looks like this:

0,         1,            1,            2,            3,            5,            8,            13,            21,            34,            55,            89,            144,            233,            377, etc., etc.

That is:



2+1 (the “number” before)=3

3+2 (the number before)=5








etc., etc., etc.

In Spiritual Theory, Life must look back on itself before it can move forward.  In Relation to the human species, we must look back toward our Ancestors to learn Wisdom and give Gratitude to Life in the Present in order to move into the Future in the best way and in the Best Direction.  Because in theory the Spiral is not quantifiable in the concrete sense; i.e. it is a sequence that is Infinite (no final end number), mathematicians use straight lines around the spirals to give it as close to a concrete geometric equation as is possible.  Hence, the spiral looks like a spiral of expanding squares as shown here, and is known as the Golden Mean Ratio.  In biological settings, The Fibonacci Sequence can be seen in the Spirals of the Pinecone, in the Branch growth pattern of trees, the mini-fruit pieces of the Pineapple, the Artichoke flower, a Fern during its uncurling, and Seashells.  The sequence can also be seen in Rabbit breeding patterns, and the family tree of Honeybees.

In relation to Rabbits, Fibbonaci posed this question during the middle ages:

Rabbits can mate at the age of one month, a pretty fast breeding cycle for an animal. If a rabbit population was ideal (though not biologically realistic), if one assumed that a new pair of rabbits (one male and one female) were in a field, and after the end of one month, that pair had another pair, and every pair had another pair, how many pairs would there be after one year?

End of First Month:  1 New Pair

End of Second Month:  2 New Pairs

End of Third Month: The Original Female makes a Second Pair, Equaling 3 Pairs in the field

End of The Fourth Month:  First Female makes a Third Pair, The Female born in the Second Month makes Her First Pair, etc., Now Equaling 5 Pairs.

End of “n”th Month, Number of Pairs = The Number of New Pairs (= Number of Pairs in Month “n” – 2 + Number of Pairs alive in Previous Month “n” -1.

This is the “n”th Fibonacci Number, and it looks like this:

There are numerous other examples in nature shown in this site, as well as in class activities you can do to demonstrate the Fibonacci.

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