Fundamental Form-s

[Home] [Other Creations] [Zigzag bookshelves] [Thinkertoy Swing] [Architecture] [Crystal Dome] [Crystal Pod] [Pole/Stucco/Foam SIP] [Hex-Mod Buildings] [7-sided Dome] [Diamond Pod] [Saddle-curve structures] [Fundamental Form] [Professor Fundamental] [Testimonials] [Energy Breakthroughs] [Energy from Waste] [DOE report-1] [DOE Report-2] [Emissions test] [Biomass Gasifier Breakthrough] [Hot-air gasifier] [Involute Wind Turbine] [Wind Power Analysis] [inflatable VAWT] [Radical Wind-Turbine Boat] [Water Energy] [Hyperboloid Venturi] [high-pressure fan] [Writings] [Education] [PLANS] [VAWT Plans] [Low Head Water Turbine] [Other Turbines] [Water Turbine Plans] [Pole-SIP Plans] [Elijah] [Hex-Shed] [Stilt Plans] [bookshelf plans] [Contact] [Opportunities] [Wind Turbine Ops]



Let's Build a New School


Thoughts on Radical Education




The following essay by Lawrence Dobson is a beginning attempt to voice the dream of establishing
a radically new learning environment on South Whidbey Island, Washington. {written 1974}


I. The Present System

America has evolved into the most materialistic country in the world. We are in love with the material world. We consume more of it than any other  culture by far. We have conquered and subdue it. Having been seduced by  transient material desires, we now awake to find we have raped, almost  killed, our priceless natural heritage. Now we repent and vow to  establish a new harmony with nature, for reverence for nature runs deep  in marrow; our deepest gratitude and joy is released through working  with her forms and energies, discovering her laws, and dancing to the  pulse of her seasons.

Physical involvement with the material world, "wading in and getting our hands  dirty", has been a typically American trait, yet in our schools we learn to talk about, rather than do. This country was built by a pragmatic  "do it now" approach, working with what was there and needed to be done. Without blind tradition to aide us, we developed a simple, direct  "horse sense". Yet an overly intellectual approach to learning in our  schools has usurped control from common sense. Concerns for following  prescribed forms and assuming legal responsibility has made it easiest  to intellectualize on a subject, rather than get physically involved  with it.

Public education has developed along one general pattern of growth, which  graduates a citizen adapted to survive in a specialized environment --  the institutional, status-quo, paperwork- oriented, conservative  American bureaucracy. The institution is so highly developed that it can no more evolve into a radically different system than the dinosaur  could evolve into the monkey. Yet, just as nature must evolve new life  forms to express her dynamic purpose, so must public education take on  new life forms that we may realize our true destiny.

In our schools the practice of damming the natural flow of childish  enthusiasm by abruptly changing focus every 55 minutes with the ring of a bell, or, on the other hand, demanding mental concentration on a  subject foreign to the spirit of the moment, may have created the  analytical detachment necessary to focus critically on our present  dilemma, but it leaves us without the practical skills to resolve it. If we now have the vision, we seem powerless to act on it. Having evolved  from a nation of pragmatic doers into a society of servile sitters, we  are ruled by regulations that quickly exhaust the energy released for  active involvement .

The public school system is self-perpetuating and non-corrective. As it  becomes larger, it becomes top-heavy, so that often less than half of  the employees are teachers. The desire by those out of contact with the  students to know what is happening creates time-consuming accounting,  regulations that are unreal to the students, classes of unruly mobs, and impersonal distance between authority and subject.

Learning must proceed from the students and flow along their channels of  interest. The teacher's job is to strengthen the flow, remove obstacles  from the path, and entice enthusiasm to branch out into the many fields  of human delight. Excessive control from above sucks energy from the  system, which can only be captured by damming the natural flow of  bubbling enthusiasm to create a quiet, passive, artificial mind that is  slowly silting up. What happens when the dam breaks? Authorities shudder at the thought, and quickly add another regulation.

The school system cannot change from within; it lacks the integral vision.  It may produce well-disciplined, law-abiding citizens, but what  treachery to turn them loose into a society ruled by leaders without  vision and laws that have lost their meaning in the maze.


We must begin a radically new orientation to education, even if it is not  appropriate yet for the majority. Radical is defined as, "arising from  or going to a root or source; fundamental; basic." In mathematics, a  radical number can he approached by a fraction or decimal, but it can  never be exactly pinned down; it is an irrational number.  (The  circumference of a circle, pi, the diagonal of the square, the square  root of 2, he diagonal of the cube, the square root of 3 are all radical numbers, quite real to our perception but unmeasurable.) This is  the problem we face in trying to define fundamental, basic education,  and, how totally different it is from the preprogrammed education we are used to. Fortunately, just as an irrational radical number can he seen  to be as real as a rational number, radical education can be seen to be a real alternative to our present system, which reason might convince us  can be modified to fulfill our needs. The path of warring duality in  denial can never lead to integrated, harmonious self-discovery.

CUBE_MERKABA202An analogy may clarify this point. Let the institutional view of basic  education be represented by the square, the shape we are most familiar  with, each side measuring one unit. (diagram above) This is the right  shape to begin with, we reason, since one is the most elementary number, and all the angles are "right", 90, the angle we make with the surface of the earth when balancing against gravity. We measure our world with  this angle; front-back, left-right, up-down; length, width, height; all  at right angles to each other. If we slice space with three planes at  right angles to each other, assuming that any slice parallel to another  is the same slice, since they never intersect and must be therefore be  connected at infinity. This means we can define every point in our  three-dimensional world by the cubeic matrix in space.  Therefore,  we think, this must be the proper shape to build our institutions of  learning. But when we try to construct our square environment in the  real world, we discover that our framework collapses unless we add at  least one diagonal brace on each side, which length is 2, a radical, irrational number, so defined because we cannot measure it  exactly with our square ruler. But we can measure it accurately enough  to build with, so we finish building our school, write into our building codes that all buildings must have diagonal braces, and proceed to  educate our pupils with knowledge assembled from the four corners of the world.

But we have overlooked a fundamental fact that is now outside the knowledge of our system. The six diagonals of the six sides of our unstable cube  form a simpler structure, which stands rigid by itself. It has four  sides instead of six, four corners instead of eight, and six edges  instead of twelve! But alas, from corner to corner it measures  1.4142135......etc., etc., an irrational, impossible number to work  with!  Besides, who can even pronounce its name, tetrahedron, and  how confusing to think of it in terms of length, width, and height! So  we can approach an understanding of the tetrahedron, but never clearly  see it, for to do so would necessitate enthroning the tetrahedron as our one ruler (which would make the square world irrational, measuring 2/2), and thinking in four dimensions, rather than three. We would have to  develop a whole new way of thinking, based on this new conceptual  framework and "irrational" system of logic. The world would look as  different as it does through the eyes of an Indian astronomer compared  to an American psychiatrist. But it would be as real a world as the  square, three-dimensional one, perhaps even more so, since when we  account for the two possible diagonals of the cubic sides, we get a dual tetrahedron, which is the foundation of the energy matrix which  determines the structure of our crystal world of atoms! Just as  admitting irrational numbers into the world of real numbers has opened  new vistas of mathematical discovery; so there is now the need of an  "irrational" approach to education to give our nation a balanced vision  and direction.

An individual with a predilection toward seeing from as fundamentally  different a perspective as our example would find the public school  environment alien to his nature, for public education in America has  long had the job of homogenizing diverse cultural elements into a  cultural identity. The emphasis has been, consciously or unconsciously,  on developing a standardized American perspective, discouraging radical  diversity of thought. Now we are so thoroughly Americanized that a wave  of discontent and disillusionment is sweeping over the land. We open our eyes to see that unity has come to mean stifling uniformity, and justice has become the slave of mediocrity, for all the ruling elite. What has happened to the invigorating delight of  diversity, the freedom to be different, for which we have shed the blood of countless thousands?  It lies dormant in all of us, crying for  release from the cultural, institutional mold .

Deep within all of us we feel a uniqueness, that if we could but fully  express it, would make us so different from anyone else that there would be no competition, no jealousy, no hatred; only a harmonious play of  such vitality and strength of diversity that it would truly be heaven on earth. To nurture this divine personality should be the aspiration of  our school.

We need an alternative fundamentally different and divorced from the  present system. Otherwise we will never be able to see the forest from  the trees. It must be deeply rooted in the noblest aspirations of all  our ancestors, back before our blood was joined on this soil. It must be universal in scope, uniquely American, but exclusively designed for  this particular community. To the extent it is founded on the principles that built this nation, it must to the same degree be free from the  constrictions that are strangling it. This requires the good will of the community, which can best be acquired by focusing on positive concrete  proposals, not reacting against the present school system. If it is  radically different, it will present fewer territorial challenges to the present system and trigger fewer built-in reaction patterns, provided  it is founded on principles revered by the majority of us.

Education is a process of bringing out from within an awareness of the forces,  forms and laws of being that exist in this universe of ours. Before a  mental consciousness can evolve, there must proceed a physical and vital involvement. Otherwise a blind knowledge, without personal verification, develops,  and this leads to distortions of all kinds. So our approach to education must be experientially based, both with the physical senses and with  the vital emotions. For example, the experience of swinging through the  air at a tempo determined by the length of a trapeze rope, the force of  gravity, the arch of the body, to intercept the connecting rhythm of  another swinging trapeze rope of different Lengths, provides a  familiarity with the law of motion of a pendulum such that the truth of  the equation t = 2*pi*1/g is immediately recognized, rather than memorized when formal study of such laws has progressed to that degree of abstraction.

Simply stated, the richness and fullness of life can only be fully appreciated by living it with every nerve and fiber of our body, not just in the  cool elegance of our thoughts. Alan Holden, a renowned  physicist-mathematician, stresses this in the preface to his book, Shapes, Space, and Symmetry, "Space provides no three-dimensional blackboard. We learn about space only by  living in it. A child climbing in his jungle gym may learn more about it than he will ever learn again, for his books will be made of  2-dimensional sheets of paper."

First and foremost our school must be a living experience, involving the  whole being in a rich interaction of interests from sports to academics; social, sexual, cultural, creative, artistic, spiritual pursuits. This  sounds like the ideal description of our preferred life. Why the need  for a special school environment? The following discussion will focus on one answer to this multifaceted question: the need to isolate and  simplify in order to focus upon the basic laws of life's overwhelmingly  complex expressions. Our school should be an archetypal setting, where  teacher and pupil can establish basic rhythms, play with the elementary  laws of the universe, pursue their emotions and energies with guidance,  build insulated cocoons when the urge takes them inward, and pursue in  concrete form their most treasured fantasies.

Let us pursue a fantasy now, and picture a possible form of "elementary"  school environment, from the perspective of mathematics and physics. Our universe is composed of energy in definite patterns of space and time.  Let our school structures express these archetypal contours. We are not  familiar with the shape of atoms in crystal array, the harmonious  pattern of their interactions, yet most of the world around us is built  upon these few fundamental relationships. These forms would make elegant buildings, providing the potential for an integrated modular building  complex of infinite variety, and moving within their patterns would  awaken a sense of the universal rhythms of life in mater.

We are quite familiar with the cube, so exclusively familiar, in fact,  that its three dimensions we consider to be at "right" angles to each  other. All other angles are felt to be less perfect, not "right",  despite the fact that the "right" angle for the atomic structure of the  majority of the earth is 109 27' 16....", a radical, unfamiliar angle.  Gravity forces us to relate to the earth at right angles, thus it seems  appropriate that the ancients should have seen the cube as the elemental form of the earth. But they knew other elements and other elementary  forms: Earth, Air, Fire, Water, and Ether are expressed in pure form as  the Cube, Octahedron, Tetrahedron, Icosahedron, and Pentagonal  Dodecahedron, respectively.

          the All Five Puzzle02 

These elementary shapes, with every edge and angle identical, are truly  archetypal, for they are the five simplest (regular) vibrations of  space. Plato was so fascinated with this truth that they are commonly  referred to as the Platonic Solids.

What difference does the shape or orientation of a structure have on the  consciousness that develops within it? Acoustically, the cube has the  greatest echo, because there are two opposite walls to reverberate in  each of its three dimensions. The cube has the greatest reflective  symmetry, expressing duality in the highest degree. No wonder it is so  familiar to us! The tetrahedron, fire, simplest of all form, has no  reflective symmetry. Who is familiar with tetrahedral space? Surely we  would feel different in a bare room that was as 'live' as a cubic  handball court, but with no direct echo. Shouldn't we be experimenting  with these elemental forms?

We have learned to focus, filter and direct light through physical matter  in the shape of lenses, prisms, mirrors, crystals. Other frequencies of  vibration close to light we are learning to control, such as heat, radio waves, x-rays, microwaves, etc.. But existence expresses itself as an  infinite variation of vibrations, and we must have the potential for  awareness and control on levels undreamed of by most of us blind souls.  If we can put together materials of a particular atomic shape into a  certain relationship, and focus our familiar vibrations, why not  likewise for these other octaves of awareness? Research into the Great  Pyramid of Cheops and many deliberately shaped and oriented structures  and lines of communication throughout the world overwhelmingly suggest  that the ancients knew how to control fields of energy unfamiliar to us  outside of our fables.

Let us pursue this quest for ever-expanding sensitivity and control by  organizing our school environment based on these teachings, from the  Great Pyramids and Chinese geomancers to Theodore Reich, Victor  Shauberger, Buckminster Fuller, and a host of great teachers of all  persuasions. History has placed in our hands the collected knowledge of all the peoples of the world. Our task is to thoroughly enjoy the  diversity and the unity. We know not how, until we have experimented  with it. We know not when, beyond our creative presence, where it  arrives in perfect step.  The physical school environment is a  logical place to begin.

There are simple mathematical curves that represent fundamental flow patterns of energy in our world. Let us build our playground from such patterns. Gravity, for example, flows most elegantly along certain curves. A ball rolling down a track in the shape of a catenary curve will reach the  bottom quicker and be going faster than on any other shaped track; the  tractrix curve changes direction smoothest of any curve; and the  parabola is the path taken naturally by a projectile under the influence of gravity. Let us combine these curves to make a "roller coaster"  where children can experience these 'truths totally, integrally, The  supporting framework could graphically demonstrate how these curves are  generated mathematically.

If the student is to develop a balanced sense of the flows of energy  within his/her environment, the school should be as self-sufficient as  possible. How much energy is used to heat the buildings? If this energy  comes from outside, from an oil truck or power pole, the student never  really learns that reality. If part of his/her education is dealing  directly with this concern, she/he will learn the balances of ecology,  learn conservation of energy and establish the basis for a simpler, more direct life-style. Let us incorporate into the school solar heating and lighting, windmills, wind turbines and water pumps, rainwater  collection and storage systems, various methods of power storage, such  as batteries, electro-gyroscope, elevated water reservoir, rock-,  water-, earth- heat storage systems, electrolysis of water and use of  the hydrogen/oxygen for welding, lighting, etc. Students could have  numerous jobs monitoring the equipment, where they would become aware of the cycles of the sun and wind, and the translations of energy forms  one into another. With such a familiarity with energy systems, the laws  of physics and chemistry should appear much more obvious and practical.

Wood is an important materials in our life, yet we are blindly consuming our forests with little awareness of forestry management and the many  properties of wood. Let a tree farm be part of our school, with students actively involved in cutting firewood, logging, replanting new species, developing park areas, and keeping track of the life of the tree farm.  They could graft alder archways for blackberry farms, grow ornate  furniture and all sorts of trained tree creations. Lumber for new  buildings and furniture could be milled and seasoned by the students.  They would develop a greater reverence for wood and the objects made  with it, as well as a greater knowledge of the material and how best to  use it.

Food is one of our most basic needs and should certainly be a primary focus  of an enlightened education. It would seem appropriate to have a school  farm, where all the foodstuffs of a balanced diet could be cultivated.  Here, knowledge of local weather, soil, natives, particularly  well-adapted species of plants and animals, and the collected experience of local farmers could be assembled, further experimented on, and that  accumulated knowledge made available to the larger community.

In a traditional school environment, where everything closes down at 4  o'clock, such a farm, especially where animals are involved, is  impractical. Yet, if our school is to offer a real education for life in its unpredictable fullness, we cannot turn it off and on like a TV set, so we must expand our vision. The radical picture that comes to mind is a community of boys, girls, men, women, babies, pets and barnyard  animals, many living "on campus" for periods of time, pursuing numerous  ongoing projects for internal development, economic venture, "pure  research" and pure fun. Various trades and skills would be practiced,  with emphasis on perfection of the art of living by the harmonious  integration of the necessities and fascinations of life.

All actions would be guided by the goal of establishing a radically new and universal Earth Consciousness of loving grace, and creating the fertile environment for its growth.

The following is a course outline for my teachings in FUNdamental Geometry (always changing!):


    Session one

  • Our 3-dimensional world: how we perceive it.
    • an etymological perspective
    • normal, right, real, orthodox, radical, dimension, etc.
  • The Cube as the foundation for 2-dimensional thinking
    • the cube inside-out: the rhombic dodecahedron
    • the face-centered-cubic crystal structure: different ways of perceiving space
  • The tetrahedron within the cube - a more fundamental form
    • The diamond bond and the true diamond shape: the building blocks of both organic and inorganic world
    • a class shape-preference pole

    Session two

  • The diamond as the foundation for 3-dimensional thinking
  • seeing cubes in the diamond matrix
  • garnet
  • honeycomb
  • stellated rhombic dodecahedron
  • moving from one space grid to another
  • square-root of 2 & 3 as primary proportions
  • atomic refraction photos
  • Class construction project
  • Session three

  • Space-filling shapes
  • tessellations in 3-D
  • cube
  • rhombic dodecahedron
  • tetrahedron with octahedron
    • Buckminster Fuller’s Tensegrity structure, the Vector Equilibrium
  • Why there are only 92 natural elements
  • playing with blocks & puzzles
  • discover the body-centered-cubic crystal structure
  • discover the diamond crystal atomic shape
  • class project---building a space grid
  • Session four

  • Atoms as spheres
  • playing with marbles, building crystal arrays
  • seeing space as interfacing energy waves or as interpenetrating energy grids
  • dual forms and dual grids
  • playing with soap bubbles and soap films
  • the diamond bond angle as a universal relationship of balanced proportion
  • soap film curves
  • hyperboloids, hyperbolic paraboloids, curves of all straight lines, and curves of zero curvature
    • interconnecting curved planes in elementary space grids to form interpenetrating space volumes
  • make your own unique curved space matrix
  • Session five

  • The five Platonic Solids
  • their mathematical and mystical significance
  • Volume ratios based on the tetrahedron
  • The three geometries of the world of matter
  • The two geometries of Life forms
  • viruses, pollen, spores, radiolaria, flowers, etc.
  • fundamental reproductive differences of the two geometry classes
  • Playing with blocks and construction sets
  • Session six
  • The Divine Proportion:
  • The Fibonacci series and the five-pointed star
  • 5 as the basis of  magical interrelationships of branching patterns in nature
  • class show and tell
  • Session seven

  • The Great Pyramid of Cheops as a synthesis of  and
  • The magical number 25,920
  • The procession of the equinoxes
  • rhythms of the human organism
  • earth's size and gravity
  • The concept of "Universal Energy Matrix"
  • The Earth is a giant crystal
  • The Russian hypothesis
  • power spots, weather fronts, natural energy grids on Earth
  • Session eight

  • Fundamental curves and their properties
  • The conic sections, circle, ellipse, parabola, hyperbola
    • elliptical, parabolic & hyperbolic reflectors
  • the catenary curve and its relationship to the parabola
  • bridges and spinning buckets
  • the tractrix
  • freeways and bearings and grinding stones
  • how they all interrelate
  • curves for windmills & gears
  • Session nine

  • Putting it all together
  • architectural applications
  • class project: building a campus of the future with geometric blocks, construction sets, and student-designed structures.
  • Session ten

  • Class project, continued
  • student and teacher reports on relevant topics of general interest

FUNdamental Geometry ~ the Geometry of Natural Form

Preliminary Course Description

Discover the elegantly simple and magically interrelated geometries underlying  nature. Learn the language of Form fundamental to Science, Art,  Engineering, biology, etc..  Explore the interrelationships of  crystal structures, life forms, space grids, the Platonic Solids, the  Great Pyramid, bridges, bubbles, honeycombs, sunflowers, sound  reflectors, the Divine proportion, curves & curved space, etc. Play with geometric blocks, puzzles, construction sets, marbles, soap  bubbles and films. Build a new world of archetypal forms with straws  & cardboard, etc.

Geometry of Natural Form has been a passion of mine for over 40 years.  I have  come to realize how elegantly simple and aesthetically beautiful are the underlying forms and principles upon which nature builds to create  everything from atoms to honeycombs.  I have constructed many  models, blocks, puzzles and construction sets to illustrate the magical  interrelationships of elementary forms, and when I show them to people,  they too get excited and often comment that they wished mathematics had  been as fascinating and understandable in school. I feel that this  subject should be part of basic education and could be introduced in  elementary school with blocks and construction sets. 

The geometry of natural form is fundamental to many fields of  interest.  The engineer will find the strongest, most modular and  most easily analyzed structures in the crystal world.  The chemist, electronics engineer, geometrician, biologist, botanist, etc. will see  their fields magically intertwined at their roots.  The artist and  metaphysician will delight in the infinite simplicity of archetypal  forms that interpenetrate, transform, and reappear continually in  nature's intricate dance.

A sense of order is essential to everyone's stability and sanity.   An awesomely simple order underlies our complex physical world, and in  its recognition lies the potential for liberation from our flat  superficial world-view. The student should leave this course  with an elementary mastery of an archetypal design language from which  to understand the universe and evolve his/her own unique forms of  expression.

Students will be continually challenged to see the myriad interrelationships  between elementary forms. They will make models of straws and paper and play with construction sets, geometric blocks and puzzles. students  will need pencil, note paper, ruler, compass, protractor, and colored  markers.  They will also be asked to bring to class examples of  flowers, honeycombs, crystals, etc. that they have ready access to  without cost.

 I have taught classes on this subject at Tomanhaus School and Community  School on South Whidbey Island; at Evergreen School for the Gifted, and  at The Northwest School in Seattle. I have built scores of geometric  models, puzzles and construction sets, geometric playground equipment  (featured on television), and houses and domes based on crystal facets.

8. Describe your teaching/training background.   I taught 4 years in Peace Corps, India, and Peace Corps training programs in the U.S.; 3 years mathematics, film making, and tumbling at  Tomanhaus School on Whidbey Island; and Geometry of natural form at 4 other  public & private schools. I have further communications experience in acting.