Mystical to Mathematical Beauty. I traced the transition from mystical to mathematical beauty in American thought: from the theologian Jonathan Edwards in the 18th century, through natural philosopher David H. Thoreau's "Walden" in the 19th, to the mathematician, Benoit Mandelbrot's "Fractal Geometry of Nature" in the 20th century. Chapter 4 of Paul H. Carr's "Beauty in Science and Spirit,"

1. Ice Melting on Walden Pond near the site of Thoreau’s Cabin
“Water Indeed Reflects Heaven” Thoreau
From Theology to Fractals
Mystical to Mathematical Beauty
By Paul H. Carr, next to
Henry D. Thoreau’s statue.
www.MirrorOnNature.org

2. 18TH Century Theologian of Nature, Preacher,
Jonathan Edwards (1703-1758) Northampton, MA
- Nature was a manifestation of the beautifying activity of God.
19th Century Naturalist, Natural Theologian,
Henry David Thoreau (1817-1862):PIVOTAL, Concord, MA
-“My profession is to be always on the alert to find God in nature,
to know his lurking places, to attend to the oratorios,
the operas, in nature.”
20th Century Mathematician Benoit Mandelbrot’s
“Fractal Geometry of Nature” (1924 - 2010 ).
IBM Research Laboratory, New York
Uncovered the mathematical structure of nature’s beauty.
From MYSTICAL to MATHEMATICAL BEAUTY
New England Thought

3. Beauty: From THEOLOGY TO FRACALS:
American Thought
18TH Century Theologian of Nature, Pastor, Jonathan Edwards (1703-1758):
- Nature was a manifestation of the beautifying activity of God.
- Beauty originated in God who communicated various degrees of
“excellency” in creation.
- The degree of beauty is determined from the rightness and
excellency of their proportions.
19th Century Naturalist, Natural Theologian, Henry David Thoreau (1817-1862):
-“My profession is to be always on the alert to find God in nature, to
know his lurking places, to attend to the oratorios, the operas, in
nature.”
- Believed Darwinian evolution was a ”constant new creation.”
20th Century Mathematician Benoit Mandelbrot’s Fractals (1924 -2010 )
The Fractal Geometry of Nature.
-The branching of our lungs and of plants are characterized by
fractal scaling (each branching generation a fraction of the previous)
- Discovered snowflakes, mountains, clouds, have fractal edges.

4. Thoreau's career: a microcosm of a macrocosm.
His career parallels the trend of science to emerge from a mystical
understanding of the beauty of nature (macrocosm) to the
mathematical.
Science had its origins in the beautiful myths of antiquity, which used to
explain the natural world.
• The Sun God Helios’ Chariot Traveled East to West each day. (Mystical
Beauty, but not good for predictions. )
• Pythagoras, 590 BC
The Pythagorean mystical sect discovered the mathematical beauty of
nature and music. They believed that the planets orbiting the earth made the
beautiful Music of the Spheres, because the ratios of the planetary orbits
were the same as the intervals of the musical scale.
Pythagorean theorem
for 90 deg. Triangle.

5. The transition from the mystical to the mathematical is evident in the
increasing accuracy for predicting the motion of the planets, as
exemplified by:
• Ptolemy, 200 AD, geocentric model of solar system predicted planetary
motion.
• Copernicus, 1543, heliocentric model was not more accurate, but more
beautiful, simpler, greater mathematical elegance.
• Newton, 1670, gravitational force & law of motion, astrophysics
• Einstein, gravity curves space-time that tells matter how to move.
(Einstein’s Special Relativity. E = mc2 c2 = E/m )
• Tegmark’s Our Mathematical Universe, analyzed the “fossil radiation”
noise form the Big Bang beginning. I call it the “Whispering Cosmos.”
Ref.Chapter 3, “From the Music of the Spheres to the Big Bang’s Whisper”

6. American Scientist, Sept – Oct 2020, pg 311
From Pythagoras to Einstein.

7. THOREAU’S CAREER:
from the Mystical to the Scientific.
“I am a mystic, transcendentalist, and natural philosopher to boot.”
Thoreaus’ mystical and philosophical
descriptions of nature in the River and
Walden:Life in the Woods were written at the
beginning of his career.
The following photos have Thoreau quotes.

8. "Color stands for all ripeness and success."
Thoreau

9. "Water, by reason of its transparency and limpidness, is the mirror of bodies -
of physical etres, so also is truth equally the mirror of ideas." (Thoreau)
Upper Baker Pond, NH

10. "A man's life should be as fresh as a river.
It should be the same channel, but new water every instant." (Thoreau)
Connecticut River, Orford, NH

11. A Week on the Concord and Merrimack Rivers
In the "Sunday" chapter, of Thoreau’s first book described how, in reading a
work on agriculture, he was annoyed at having to read moral reflections on the
“personality of God,” with which he did not agree. Thoreau wrote:
“There is more religion in man’s science than
there is science in their religion…
“A man’s real faith is never contained in his creed.”
Thoreau’s argument against the creeds of organized religion reflects the
beliefs of his fellow transcendentalists, who wished to restore the importance of
individual inspiration. The transcendentalists felt that the creeds of organized
religion were burdensome anchors, which prevented the creativity of individual
revelations and epiphanies.

12. Paul with daughter & grandson. Thoreau Society Annual Gathering, July 12, 2016
See https://www.thoreausociety.org

13. “The streets of the village are much more interesting to me at
this hour of a summer evening than by day. Neighbors, and
also farmers, come a-shopping after their day’s haying, are
chatting in the streets, and I hear the sounds of many musical
instruments, and the singing from various houses. For a short
hour or two, the inhabitants are sensibly employed. The
evening is devoted to poetry, such as the villagers can
appreciate.”
HENRY DAVID THOREAU (1817-1862)
Journal Account of a Summer Evening
in Concord, July 21, 1851 at 8:30 P.M:
During the Thoreau Society Annual Gathering, our walking-tour guide read this
passage while walking us through the streets of Concord, MA,

14. You do not have to be good.
You do not have to walk on your knees
for a hundred miles through the desert repenting.
You only have to let the soft animal of your body
love what it loves.
Tell me about despair, yours, and I will tell you mine.
Meanwhile the world goes on.
Meanwhile the sun and the clear pebbles of the rain
are moving across the landscapes,
over the prairies and the deep trees,
the mountains and the rivers.
Meanwhile the wild geese, high in the clean blue air,
are heading home again.
Whoever you are, no matter how lonely,
the world offers itself to your imagination,
calls to you like the wild geese, harsh and exciting -
over and over announcing your place
in the family of things.
WILD GEESE
by Mary Oliver (1935 -2019)
After an abusive childhood, she was
“saved by the beauty of the world.”

15. Geese Flying Over Melting Ice, 29 January 2002
"I look into the placid reflecting water for the signs and promise of the morrow.”

16. THOREAU: From MYSTICAL to MATHEMATICAL BEAUTY
“Mystic, Transcendentalist, and Natural Philosopher” (Scientist)

17. THOREAU’S CAREER:
from the Mystical to the Scientific.
“I am a mystic, transcendentalist, and natural philosopher to boot.”
His mystical and philosophical descriptions of nature in the
River and Walden:Life in the Woods were written at the
beginning of his career.
The rest of his life was spent in scientific observations,
such as the blooming time of flowers described in his
journal. His interest had shifted from the human world to the
natural world, from the cultivated fields to the wild woods.
In Walden, the agriculture of his bean field is a metaphor
for human self-cultivation.
In his later journals, Faith in a Seed, & Wild Fruits, wind,
water, and animals are agents for nature's self-cultivation.

18. SPONTANEOUS GENERATON?
"I have great faith in a seed. Convince me that you have a seed
there, and I am prepared to expect miracles.“ Thoreau

19. Thoreau read Darwin's Origin of the Species shortly after its publication in
1859 and wrote,
"The development theory (evolution) implies a greater vital
force in nature, because it is more flexible and
accommodating, and equivalent to a sort of constant new
creation."
Thoreau was thus one of the first Americans to accept Darwin's theory, in
contrast to the Harvard's renowned Professor Agassiz,
who believed that the geographical distribution of species was
"regulated by the limits marked out on the first day of creation.“
“The most distinct and beautiful statement of any truth must take
at last the mathematical form.” Henry David Thoreau
THOREAU: AS SCIENTIST
“I am a mystic, transcendentalist, and natural philosopher to boot.”

20. THOREAU, A PIVOTAL FIGURE:
Ahead of his time.
18TH Century Theologian of Nature, Preacher, Jonathan Edwards:
- Nature was a manifestation of the beautifying activity of God.
- Beauty originated in God who communicated various degrees of
“excellency” in creation.
- The degree of beauty is determined from the rightness and
excellency of their proportions.
19th Century Naturalist, Natural Theologian, Henry David Thoreau:
-"The development theory (evolution) implies a greater vital force in
nature, because it is more flexible and accommodating, and
equivalent to a sort of constant new creation."
-“The most distinct and beautiful statement of any truth must take
at last the mathematical form.”
20th Century Mathematician Benoit Mandelbrot’s Fractals
- Authored The Fractal Geometry of Nature
- The fractal beauty of nature, like evolution, is characterized by the
interplay of chance & necessity, variations & the natural selection law.

21. Mathematician
Benoit Mandelbrot
(1924 -2010)
Author of.
The Fractal
Geometry of Nature
speaking at TED
conference.

22. Sierpinski's (1882-1969) Fractal Triangle 13th Century Art
Artistic Premonition of Math?
Pg 43 of Stephen Wolfram’s “A New Kind of Science.”
used with permission.
Mathematical from Mystical Artistic
Beauty

23. Fractal Branching of Trees and Leaves
Each generation of branching is a fraction of the previous

24. The lower leaf, enlarged from the fern on top, shows similar fractal
branching.

25. Black dots of FRACTAL TRIANGLE generated by:
(1) Random initial point & direction,
(2) Deterministic Rule: Go 1/2 distance to vortex and draw
a point.
from Edgar E. Peters, FRACTAL MARKET ANALYSIS: Applying Chaos Theory to Investment &
Economics.
• Randomness in the algorithm constrained by
determinism results in orderly object.
• Analogous to Darwinian evolution by random
variations & deterministic natural selection.

26. Mystical beauty is difficult to define,
but you know it when you experience it.
Fractal Mountain Landscape with
Mystical & Mathematical Beauty
Richard Voss generated this landscape with a computer program having
both random and deterministic elements. Used with permission.

27. OLD METAPHORS:
“God does not play dice.” Albert
Einstein
“I do not believe that God could have
created us in His Image by mutations of
the genes.” Huston Smith
NEW METAPHOR:
“God plays dice, but the dice are loaded.”
Chaos theorist, Joseph Ford

28. THE DICE
ARE LOADED
If these natural
phenomena
were
completely
random,
K would be 0.5

29. THE DICE ARE LOADED
K greater than 0.5 means that
it is more probable to have
seven years of plenty followed
by seven years of drought,
than if K = 0.5, (for total
randomness).
(Story of Joseph in the Bible).

30. 0.5 complete
randomness.
from E. E. Peters, FRACTAL MARKET ANALYSIS: Applying Chaos Theory to Investment & Economics.
If we think of risk
as standard
deviation, risk
decreases after
1000 days (about
3 years).
Long term
investors incur
less risk than
short term. i.e.
Buy and Hold
Strategy.

31. From THEOLOGY to FRACTALS
Mystical to Mathematical Beauty
18TH Century Theologian of Nature, Preacher, Jonathan Edwards:
- Nature was a manifestation of the beautifying activity of God.
- Beauty originated in God who communicated various degrees of
“excellency” in creation.
- The degree of beauty is determined from the rightness and
excellency of their proportions.
19th Century Naturalist, Natural Theologian, Henry David Thoreau:
-"The development theory (evolution) implies a greater vital force in
nature, because it is more flexible and accommodating, and
equivalent to a sort of constant new creation."
-“The most distinct and beautiful statement of any truth must take
at last the mathematical form.”
20th Century Mathematician Benoit Mandelbrot’s Fractals
- Authored The Fractal Geometry of Nature
- The fractal beauty of nature, like evolution, is characterized by the
interplay of chance & necessity, variations & the natural selection law.

32. Swallowtail Butterfly
with Divine Proportion 1.618
BEAUTY in
SCIENCE & SPIRIT
Chapter 4,
"Beauty:
From Theology to Fractals"
“The most distinct and
beautiful statement of any truth
must take at last the
mathematical form.”
Henry David Thoreau

33. American Scientist, Sept – Oct 2020, pg 311