Return

Cold morning garden
In the small it all returns
Big moon little moon

NG1604Above, North Healing Garden, upper terrace “Heal Forgotten Islanders” square donated by a local business. At noon o’clock, paperbush Edgeworthia Chrysantha; two o’clock daffodil stems, gift to heal Pat; three o’clock herbacious peony and flowers from Saxifrage, four o’clock, sweet bay, six o’clock, oriental poppy Moyer’s Red, a beautiful saxifrage with flower stalks forming graceful bowed arcs (gift), flowering columbine (gift), trumpet lily, snowdrops (Gilanthus nivalis) and scilla tubergeniana; seven o’clock, Moyer’s red oriental poppy; eight o’clock, checked lily; nine o’clock, marjoram.

Completion

Completion enters
In the fullness of its time
The moon wanes with pride

NG2ter1504Above, North Healing Garden, second terrace, center Yarrow “Moonbeam” to its left an Austin Rose “Marguerite”, spring onions, Moyer’s red oriental poppies, lower left corner fragrant sarcococca or sweet boxwood. The second terrace is nicknamed the “Aisle of Lilies”.

BI39Drawing: Stella Ross-Craig
Watercoloring: The Healing Garden gardener

Constructing A Heptadecagon

Note: The Healing Garden gardener’s pledge to a Healing Garden portrait each day was always intended to include a watercoloring. On our present quest for practice of our brushwork we have been reading three books.

Arnheim, Rudolf “Visual Thinking”
Byrne, Oliver (translator) “The Elements of Euclid
Sze, Mai-Mai (translator) “The Tao of Painting”

Reading “Visual Thinking” is helping me understand two types of human bias. As living human scientists exploring our world, we must choose a frame of reference.

Consciously, unconsciously and viscerally we deny certain perceptions of our experience to discover the unknown. And further educate ourselves in the business of life. There is another kind of denial, too.

It is natural to be whole. We have the gift of time so everything does not happen all at once. We accept and deny the imperfect and the darkness with which we do not have the capacity to illuminate. This denial is natural, too.

One road along the edge of denial helps us see the world more clearly. We welcome others within this circle. The other road to denial protects our incompleteness from the world and we become increasingly uncomfortable when we are robbed of these illusions untimely.

Reading “The Tao of Painting” helps me focus images dedicated to my reflections from the Healing Garden. Although these images are representational, I am not sure their invisible content is. These are my efforts to bring my thoughts to the visible. Our human culture has invented some beautifully elegant and complex branches of knowledge for communicating about our relationship to life in the world all around.

In ancient China and Japan, the use of the brush was considered essential for the educated person. Watercolors and brush and ink achieved the highest level of respect. For their beauty, and ability to express the harmonies apparent in our relations to life. Soon I will begin the First Canon in the Tao of Painting. It is about choosing and developing a style of brushwork. I am ready.

Euclid’s Elements, especially the color coded 1847 Oliver Byrne translation, provides proofs forming the basis of modern mathematics. Euclid codified the rules by which we verify our senses. In the metaphoric language of Paul Klee, this gives intent and direction to the “lines” of classic Greek geometers.

It is through the senses we apprehend the world, though we show our intuitive senses mistrust. Subsequently, rigorous rules of logic, used by the mind can discern the real world, which springs of the invisible, according to Greek mathematicians.

In my leisurely tour of Euclid, I was detoured by the construction of a 17-sided figure. A heptadecagon. Anybody can find the instructions for constructing a heptadecagon online. The proof of this construction, and its relation to traditional mathematics is an important illustration of what we humans gain and lose in the Healing Garden as we manipulate our images like tools.

Euclid’s proof [here]. Euclid’s construction proves the line segment dividing a circle into 15 equal parts. Drawn by Carlos (my fourth grade gardening helper), and by the gardener with chalk and string on the sidewalk [here]

Wentworth1504Above. Cited under Fair Use “A Textbook of Geometry” by GA Wentworth. Ginn & Company. Boston, 1898.

At the age of 19, in 1796, (2,100 years after Euclid wrote the “Elements”) Friedrich Gauss proved the construction of the heptadecagon (a 17-sided figure) was possible. One hundred years later, a generally accepted textbook of mathematics (above) proves the construction of a 15-sided figure. It would take almost another one hundred years for the world to appreciate Gauss’s geometrical discovery in 1796. Friedrich Gauss combined the geometry of Euclid with the spiral of Archimedes [here].

Stay tuned for more clues on the trail of seeking the heptadecagon.

Carlos’ Garden

Note: A few weeks ago, Carlos and his Mom approached me and asked if I would help Carlos with his vegetable plot. We have been working together each Sunday for an hour or two. [here]

CGlist21304My goal this Sunday was to help Carlos share his gardening experience; and to remind Carlos of our ongoing loyalty to his vegetable garden; and to prove, together, Proposition XVI in Euclid’s Fourth Book of Elements on our sidewalk.

For us both, concentration and attention are challenging. Carlos and I learn to compensate with our strengths. We started with a list:

(left) Carlos’ Gardening Journal

We checked on his vegetable garden. Radishes are sprouting. Radishes are good to start from seeds where one needs something quickly to demonstrate that plants grow. We strung the pyramid for peas. Carlos watered and weeded (below).

CG13042Last week Carlos, his Mom and I discussed helping a neighbor with her garden. Carlos’ neighbor is his age. Her name is V. First we talked with V’s Mom. It was a good idea to check first, because V’s Mom has other plans.

Carlos and I have been discussing a Spring garden for his Grandmother. Our Island’s second historical community garden was installed within our public housing project. Private donors generously provided a half dozen planters for the handicapped. These were not intended to engineer conflict between disabled and able bodied tenants.

Sharing is not as easy as it looks. Last week, able bodied tenants were told that the handicapped planters had no takers. A half dozen tenants in wheelchairs were told there was more demand for handicapped planters than supply. Even those like Carlos’ Grandmother who had reserved a handicapped planter were denied. Last week, I saw an able bodied gardener who does not live here, planting a handicapped planter.

CGjournal13042Some dramas are not worth the speculation. It is enough to be grateful for those who raise the threshold for our inclusion. They make us strong. Excluding us helps us improve ourselves. In the Healing Garden, excluding our neighbors is sufficient encouragement for us to help them feel included. A survey of the deep shows this is a valuable lesson for sharing the gardening experience.

Carlos prepared a site plan for new handicapped planters (above), since there are not enough for everybody who would like a garden. The plot is an irregular figure and required practice to draw. Below, new planter installed. Happy gardeners.

CGAlison1304Above. Carlos’s neighbor Alison who is mobility challenged and enjoys gardening and photography. She has a sunny spot with room for two planters. One is for her. One is for Carlos to grow flowers for his Grandmother.

Today, Carlos helped deliver and install planters for two disabled tenants in A Block. Then, we delivered the first planter to Carlos’ neighbors in C Block. It was a busy day sharing the gardening experience. Carlos will be returning to prepare another site plan and help install plants.

CGDave13042Above. Dave and his new service dog Beaulieu. Next to Carlos is Dave’s next door neighbor Candy, and her daughter. Dave and Candy have been trying to grow a garden in front of their apartments for years. Here Carlos is standing by one of their new planters.

CGPropXVI1304Expansion is likely to meet obstruction. The sun rises and at midday it begins to set. A Healing Garden gardener is resourceful in sharing the abundance of the earth. A garden is a fundamental symbol of home, and wealth and a sense of permanence in the community.

It is not easy sharing this experience. Some problems are problems because of the apparent magnitude of their difficulty. A Healing Garden gardener uses every opportunity to learn. This refines his mind. Difficulties transform themselves from obstructions into opportunities for acquiring meaning.

(Above, Carlos proof of Prop. XVI, Book IV Euclid’s Elements, of which he is very proud) Carlos and I finished a long gardening afternoon proving Euclid’s Proposition XVI from Book IV. I used a string, piece of chalk and my cane for a straight edge. Carlos has new colored pencils. I loaned him a draftsman’s compass and straight edge. I was happily surprised because he followed the logic of the proof, but has difficulty transposing shapes when the invisible becomes visible. I understand this completely.

CGPropXVI13043I did not mention to Carlos that the small line segment on the circumference between the base of the triangle and the base of the pentagon is what concerned Frederick Gauss. Carlos can discover this on his own. Our planters are circular stock tanks. I told him we wanted an equilateral triangular design fitting inside the circle. And a five sided design fitting inside the circle. Both with a common apex. We understood each other completely.

With only a compass and straight edge, in his gardening journal Carlos constructed an equilateral triangle and a pentagon within a circle and with comprehension. It’s all in the small.

Alpha And Omega

Beginning and End
We know where lines stop and start
Moonbeams gleam tonight

Hperion1304Above, North Healing Garden, lower terrace, Day Lily or Hemerocallis, “Hyperion Elite” planted last Autumn and dedicated to healing those targeted by bullies and saboteurs. Hyperion Elite is an exceptional lily for our purposes in the Healing Garden, especially since we saw the last buds (they did not flower) on December 28. This plant is a hardworking healer. The flowers are very fragrant and edible.

The Curve Of Life

Note. In the Healing Garden, we have been studying geometry as part of our “painting plants like a vegetable” brushwork practice exercises. [here]. From Euclid’s foundations we jumped ahead 2300 years to the construction of a 17-sided figure, the heptadecagon.

Euclid did not concern himself with the spiral. Less than 100 years later, Archimedes did, fortunately for Frederick Gauss and everybody else who enjoys the benefits of modern technology, mathematics and science.

We can all eyeball it. Gauss proved by rigorous mathematical logic that we are not fooling ourselves by what we see. From the Healing Garden we have seen appearances deceive. This remains unsettling each time. Appearances can be as deceiving as the intent and direction of a line (unless it is expressed with absolute clarity); and it is difficult for the gardener to determine with absolute certainty, but appearances apparently deceive everybody else, too.

Gauss needed the work of Archimedes (and Euclid, and Bernoulli, and Fibonacci) to prove the construction of a 17-sided polygon inside a circle. Frederick Gauss’s work elegantly describes the mathematical analysis of the infinite. There is some infinite in all Life. The infinite is a big subject we don’t have time to go into here. The ancient Greeks avoided infinity all together.

Spiral13041Above. The Spiral of Archimedes and below the logarithmic spiral. Cited under Fair Use from “Chaos and Fractals” Peitgen, Jurgens et al. Springer-Verlag 1992 p. 186

The spiral is important, because it is. The Healing Garden gardener suspects everything with roots in the Infinite and the Eternal is to be highly respected. The mathematical proportions in spirals can be found in everything visible in life that springs from the invisible. Frederick Gauss needed the mathematics of the equiangular spiral to pave the way for his own construction of the heptadecagon.

13042Above. The Golden Spiral whose harmonic relations are found in all the artifacts of life from leaves, to bones and anatomical structure, planetary motion and the geometry of galaxies. Cited under Fair Use from “Chaos and Fractals” Peitgen, Jurgens et al. Springer-Verlag 1992 p. 191

Attribution belongs not only to the ancient Greeks and great minds like Euclid and Archimedes and Gauss, but also to ancient India and Arabia. Regarding numbers, the Greeks had not yet invented Roman numerals. There was no zero (an empty state of affairs). No notations by powers of 10 (unlike now, millionaires and billionaires were all the same – they just owned more or less land).

Arabic numerals (the numbers with which you count your money and your blessings devotedly everyday) and the algebraic expression of your blessings increasing or decreasing over time (and the rate at which you become more or less blessed) would wait for the decline and fall of the Roman Empire, the Dark Ages to pass in Europe and the Renaissance to be reborn. With cultural permission to explore points vanishing into the infinite distance, perspective was born in Renaissance Art. Now that we have perspective, we have everything. I guess. It looks good in Renaissance Painting.

No discussion of the curve of Life, however brief, is complete without mention of Gyorgy Doczi and his amazing book, “The Power of Limits: Proportional Harmonies in Nature, Art & Architecture”. Shambala Publications. Boulder and London. 1981. Among the many awesome descriptions and overlays of the spirals everywhere in our lives, Gyorgy Doczi opens his book, below, with an overlay of the daisy flower.

Below. Under Fair Use, Page 1 “The Power of Limits” op cit

13043

 

Constructing A Heptadecagon

Note: From the Healing Garden we have been following with interest the experiments of Jack [here], in making the invisible visible. He is using the recipes of artist Paul Klee. Jack is accomplished at watercolors, and drawing, which he calls, “Taking my Line for a walk.”

Each of us finds different activities helpful when sailing above the deep diving for the solutions to problems we have not yet named. Jack was hard at work with a compass and ruler, thinking about walking a Line. He mentioned self-explanatory directions for constructing a heptadecagon (17-sided figure).

I enjoyed mathematics research before my brain injury and coma. I have not looked at my maths library for a few years. Or more. Researching the historical development of the heptadecagon, its meaning in relation to theoretical mathematics, and some applications of this information has allowed me to refocus, too

In the Healing Garden, we interpret Jack to mean he would like hints, but not the full solution right away. The Healing Garden gardener has chosen a half dozen ‘cult-classsics’ among mathematicians. The first is an unusual 1847 translation of Euclid’s Elements by Oliver Byrne, a London surveyor.

Constructing the regular polygon in a circle has far reaching applications in mathematics, sciences, art and philosophy. Because of our limited capacity, the gardener has included a snapshot of selected diagrams from these texts, permitted under Fair Use.

Example No. 1 The Elements of Euclid by Oliver Byrne (trs) Wm. Pickering, London. 1847 p. 144

HeptaD01

Above: Euclid, the reputed author of “The Elements” lived in Alexandria, Egypt about 300 BC

This weekend, five more diagrams. The final diagram is the mathematical proof stating it is possible to construct a heptadecagon (17-sided figure) in a circle. We can then demonstrate how this is accomplished without compass and ruler. With only a piece of paper.