Constructing The Universe
The Universe may be a Mystery, but it's no Secret!

This website is a look at some interests of
Michael S. Schneider
 offered for your enjoyment and education

 

Michael S. Schneider is an educator and writer who encourages a love of learning through an appreciation of mathematics, nature, art and science.


"There is a geometry of art as there is a geometry of life,
and, as the Greeks had guessed,
they happen to be the same."
-- Matila Ghyka


How to draw the oval of the Oval Office





Geometry Within A Flower


The Golden Ratio at your Dinner Table

If you're sitting around the kitchen or waiting for your meal at a restaurant (and you're mathematically minded)
you might marvel at this unexpected appearance of the famous Golden Ratio.

Gather three identical knives (or straws, breadsticks or any other identical lengths) and find the center of two of them.
Place one vertically and then put another's tip at it's middle point, so that they lie along the same horizontal bottom.
Place the tip of the third knife at the center of the second knife, and lean it down along the same bottom line.



Believe it or not, the distances along the horizontal line form the Golden Ratio,
where the whole length (A+B) relates to the large part (A) as the large part (A) relates to the small part (B)!
It's a perfect mathematical balance relating the parts to each other and to the whole by the same, unifying rule.

The Golden Ratio is the source of elegant balance throughout the universe, from atoms to galaxies.
The nearest place to notice it is the length of your "cubit" from elbow to fingertip,
where our wrist marks the Golden Ratio division:


To see the mathematical proof of this geometric "knife construction" click here.


Constructing The Universe Classroom
 in San Anselmo, California


A Classroom for adults and youngsters
Artists
Artisans
Architects
Students
Educators
Home Schoolers
UnSchoolers
and students of the geometry of life and art,
the natural sciences and philosophy.

A recent scene... exploring 3-dimensional geometry.

Which of these interconnected topics are you interested in?

Numbers and Shapes: The Universal Language of Nature and Art

The numbers 1 through 12 and the language of geometric patterns in nature, art and culture.

One and Two: Unity and Polarity
Three: Tri-Unity
Four: Fair and Square
Five: The Flag of Life (Pentagram, Fibonacci Numbers and Golden Section)
Six: The Joy of Six
Seven: The Virgin Number
Eight: Periodic Renewal
Nine: The Horizon
Ten: New Unity
Eleven: The PassageWay
Twelve: Cosmic Dozens

Beautiful Proportions in Sacred Arts, Crafts and Architecture

Worldwide traditions looked to the archetypal patterns of mathematics and nature for their compelling designs.

Natural Design and Sacred Art
The Art of Root-Rectangles
The Art of Polygons
Worldwide Healing Art
Sacred Windows
Principles of Sacred Architecture
Design A Cathedral
Design Sacred Pottery
Design Egyptian Jewelry
Golden Section Design
Islamic Patterns and Escher's Art

Constructing The Universe: Traditions of Mathematical Wisdom

The Geometer's Creation Myth
The Geometry of Fruits & Vegetables
Pythagorean Arithmetic: Figurate Numbers
The Tetraktys
Magic Squares
Knotted-Rope Geometry
The Five Platonic Solids
The Cosmological Circle
Music of the Spheres
The Spiral Path of Conflict Resolution
The Solar System Mandala
Ancient Metrology: Straightening Out Sacred Measure

Traditional Wisdom Survey

An appreciation of sacred traditions and related topics.

Egyptian Mysteries
The Eleusinian Mysteries
The School of Pythagoras
Savitri by Sri Aurobindo
Chakra Symbolism Around the World
Geometric Construction as Meditation

Click on this picture to see some geometric watercolors I've painted.

Here are some large public mandalas in Ireland influenced by my books. They're each made of natural materials and objects.

Illustrated Talk on Geometry & Egypt

I presented my findings about geometric composition in ancient Egyptian arts, crafts and architecture at the CPAK conference on Saturday October 10th, 2009 at the National Academy of Sciences' Beckman Center at UC Irvine, California.

A Golden Rectangle surrounds this Udjat eye pectoral.
Any Golden Rectangle may be divided into a square and another Golden Rectangle
by swinging an arc from the middle of a square,
as the vulture goddess Mut does here with her wing.
Similar "movements" place each element of it's design into harmonious arrangement.

Here's something interesting to ponder:

 Geometry within a painting about
the geometric creation of the universe


Many people are familiar with this illumination from the French Bible Moralisee (Vienna, ONB 2554) circa 1250.
This Bible is one of the two oldest examples of the "moralized bible" which conveyed the ideas through pictures, using the words as captions.
The picture is sometimes called "G-d The Architect Of The Universe" or simply "G-d The Geometer" and is known by other names.
It depicts Deity creating the universe as a golden sphere using a geometer's divider or "compass". Within the sphere of the living cosmos we see the World Soul.

Its source is in Proverbs 8:27 -- "When He prepared the heavens, I was there: when He set a compass upon the face of the depth..."

In Plato's Timaeus, which influenced Medieval thinkers and artists, he wrote [34b] :

"He made it smooth and even and equal on all sides from the center, a whole and perfect body compounded of perfect bodies,
And in the midst thereof He set Soul, which He stretched throughout the whole of it,
and therewith He enveloped also the exterior of its body; and as a Circle revolving in a circle...."

Among the many interesting relationships within the geometric composition of this painting
is the relationship between the two largest circles, the universe and Divine halo:

the diagonal across the square around the halo
becomes
the side of a square constructed around the (near-)circle of the universe
(red lines, below)
elegantly suggesting that an idea within Deity's mind became outwardly manifest as the universe.



But my favorite part of this remarkable painting, however, is in the detail that His foot extends beyond the frame painted in the picture,
reminding us of the mystery to where rationality and measure can never reach and paint nor words can ever go.
.

Essays on the Web

The Geometry of Rose Windows:

Cathedral of St. John The Divine (New York City)

Chartres Cathedral, France

Grace Cathedral (San Francisco, California)

The Amen Break and the Golden Ratio



The Lod Mosaic

This magnificent Roman mosaic discovered in Lod, Israel, is now on tour of the United States.
While it was designed for a flat surface, it may actually be based on a three-dimensional design.
Click on the picture to see how...

Geometry of a Kangra Painting

The Geometry of a Herter Brothers Cabinet

"Cosmic Dozens: Twelve-Fold Designs of Society and Art "

"Orienting From The Center"

See Michael on YouTube:

Egyptian Mathematics: See how the ancient Egyptians multiplied without a "times-table" the way modern computers do.

Fibonacci Numbers: See how the plant world is structured mathematically. (Introduction starts at about 2:00 minutes, Michael @ 3:00).

Red Ice Radio interview: An audio discussion about mathematics, nature, philosophy, art, science, ...

LISTEN!

Red Ice Radio of Gothenburg, Sweden, an interview with Michael by Henrik Palmgren July 11, 2010

Eric Tomb of Booktown on KVMR radio interviews Michael about numbers (May 25, 2010)

Walter Cruttenden and Geoff Patino of the Binary Research Institute interview Michael about numbers, nature, art and ancient philosophy here. (March 2008)

Hear an interview with Michael on the Voice Of America program "Our World" (27 December 2003, 5.4 Mb .mp3 audio file)

Click here to hear a review of "A Beginner's Guide To Constructing The Universe" regarding mathematics education.

Click here to hear a review (podcast or .mp3) by Jeffrey Millburn at Omni Art Salon about using the book for developing awareness.

Click here to listen to Documentary Film Maker Elizabeth Upton speak about Michael's work in an interview about "Sacred Geometry" on Mike Hagen's Radio Show (10.5 Mb .mp3 audio file).

Does 5x5=25 or 14? Are you sure?
Ma and Pa Kettle mix mathematical principles in this amusing old movie clip on YouTube.

Can 7x13=28? It doesn't seem likely, but perhaps you'll be convinced!
Abbott and Costello also mix mathematical principles in this amusing old movie clip on YouTube.

When asked, most people get this question wrong:
What do the 50 stars on the flag of the United States represent?

Most people will say that the 50 stars stand for the 50 states, but that's incorrect!
The quantity 50 does indeed represent the number of states, but the 5-pointed star is a symbol of life and humanity:
thus, the 50 stars represent the people of the 50 states, not the states themselves.
It's meant to remind us that the ultimate power resides in the hands of the people, not in any government that the people have elected.
Fifty stars represent free citizens ruling themselves, having inalienable rights, not subjects of any government deciding which rights to grant or deny. It's important to remember this.
(Although dozens of countries' flags display the five-pointed star, a few employ it this way as an archetypal symbol of humanity and others interpret it culturally in different ways.)

Is there anything interesting about the geometry of the face of the pyramid
seen on The Great Seal of the United States?
Click on the picture to find out...

A prediction from the year 1910 about education in the year 2000 --
Click on the picture for more predictions from the same source.

“Where all think alike, no one thinks very much”
-- Walter Lippmann (1889-1974)

Michael is the author of
A Beginner's Guide To Constructing The Universe:
The Mathematical Archetypes Of Nature, Art and Science
(HarperPerennial paperback 1995)
Dutch  Korean

My dear friend and teacher, the philosopher, antiquarian, artist and writer John Michell, passed into the Light peacefully on April 24, 2009.
The influence of this gentle, generous and brilliant man upon me and many others is incalculable.
Click on a picture for more...

I'm very pleased to let you know that The Book of Symbols: Reflections on Archetypal Images has been published.

(Congratulations to Ami Ronnberg and her staff for years of careful work!)

I mention it here partly because I contributed to the entry on Spirals.


Published by the Archive for Research in Archetypal Symbolism (ARAS), it's based on Carl Jung's work on the archetype and the collective unconscious.
 It's an astounding pictorial and written archive of mythological, ritualistic, and symbolic images from all over the world and from all epochs of human experience.
It contains over 800 beautiful color images, all perfectly chosen.

The book is great, an instant classic, and I highly recommend it.

If you're ever in New York City and need to research archetypal symbolism, be sure to visit ARAS at 28 East 39th Street during visiting hours and at their interesting events.

(I once gave a slide presentation and talk about spirals at one of their monthly Lunchtime Lectures.
My favorite moment at it was when I pointed out that the words "matter" and "pattern" derive from mater and pater, "mother" and "father," and the audience gasped in a collective relevation.)

About Michael ... Photo by Karen Koshgarian

Michael S. Schneider has been an educator for over 38 years. He delights in exploring the intersections of nature, science, mathematics and art.

Michael has a Bachelor of Science degree in Mathematics from the Polytechnic Institute of Brooklyn, (now The Polytechnic Institute of NYU) and a Master's Degree in Mathematics Education from the University of Florida (Gainesville). He taught youngsters for twelve years in public and private schools at the Middle School and Elementary school levels. In 1977, Michael was a Fulbright-Hayes Scholar in India studying ancient mathematics and sciences. He has been a computer consultant at the United Nations, Nickelodeon, MTV, NYTimes and many other corporations. He has worked for the New York Academy of Sciences, and wrote articles, posters and teachers' editions for various Scholastic magazines including Science World, SuperScience, DynaMath, and Teaching and Computers magazines. Michael was the creator and writer of the weekly "Mother Nature" segment at WNYC-FM radio on the popular live broadcast "Kids America" program (1986-87). He's also held workshops for educators at The Metropolitan Museum of Art in New York through their Education Department including "Science in the Art Museum", "The Mathematics of Islamic Art" and "Showing Children Harmony".In 1993 Michael worked with master stonecarver Simon Verity to design the geometry harmonizing the statues on the south side of the "Portal of Paradise" (central entrance) to the Cathedral of St. John the Divine in New York City. During 1996-97 Michael was the Dean of Mathematics and Dean of Science at The Ross School in East Hampton, NY.

Michael is the author of "A Beginner's Guide To Constructing The Universe: The Mathematical Archetypes Of Nature, Art and Science" (HarperPerennial paperback 1995), six "Constructing The Universe Activity Books" and numerous articles concerning mathematics and teaching mathematics through nature, art science and philosophy.

Although the back of A Beginner's Guide... says that Michael lives in New York City, he actually resides in Marin County, California. He's a Senior Adjunct Professor at the California College Of The Arts (San Francisco and Oakland) teaching art students "Mathematical Ideas For Artists". He's also taught at the Ex'pression College For Digital Media in Emeryville and the Sophia Center For Graduate Studies in Culture and Spirituality at Holy Names University in Oakland.