The Emptiness and Potency of 108

By Newcomb Greenleaf

Buddhist_mala_beads Buddhism and mathematics

two paths to our sacred number

maverick mathematician / vajrayana Buddhist

I see all of mathematics

as marvelous manifestation of emptiness

Mathematical objects lack

independence through three dependencies

on conventional designation

on parts and internal structure

on being parts of greater wholes

In that sense they are empty

empty of independent existence

Consider the sacred number 108

practice-etched in my psyche

Inspired by Chögyam Trungpa Rinpoche

meek perky outrageous inscrutable

crazy wisdom Tibetan yogi

with 108 bead mala I counted

seemingly infinite sequences

prostrations and refuge vows




Echoes of practice now

energize contemplation of CVIII

its emptiness

its power

For our sacred number goes by many names

many conventional designations

one hundred eight

gya gye


hundert acht


cientos ocho


cent huit

and countless more

Does 108 exist independent

of all naming and calculation?

It’s a matter of opinion

Perhaps you agree with Martin Gardner

and most mathematicians that

when two dinosaurs met two dinosaurs

there were four dinosaurs

(even though there was nobody there to count

(unless dinosaurs could count))

I algorist maverick agree only that

when two imagined dinosaurs

meet two more of same

I’ve got four dinosaurs on my mind

To paraphrase Nagarjuna

There is no mathematics without mathematicians

There are no mathematicians without mathematics

But malas have 108 beads

each journey of mind and fingers

summons 108 to life in a sacred setting

108 is a number

infinitely created

infinitely blessed

by fingers and minds

A mindful trip from 1 to 108

never skips 37

so 108 depends on 37

has no existence that is separate from 37

37 is a part of 108 that can be taken away

108 – 37 = 71

108 depends on 37 and all its parts

Count beyond 108 to larger wholes

108 becomes a part within


a billion

the host of whole numbers

These three dependencies

establish the emptiness of 108

and of all whole numbers

What distinguishes 108?

Going back 2300 years to Euclid’s Elements

every positive whole number

carries a unique structure

as a product of primes

(indecomposable numbers)

Now 108 uses only the first two primes

Holding to the symmetry

base = exponent

creates a larger pattern

the Sacred Explosion

derived from our sacred number

growing super-exponentially

Start with no primes


multiply by the prime 2 raised to power 2

then multiply by the prime 3 raised to power 3

multiply by the prime 5 to the 5th

by 7 to the 7th

by 11 to the 11th

This sixth term of the Sacred Explosion has 23 digits

Multiplying by

1313 yields the seventh term with 38 digits

by 1717 gives the eighth term with 59 digits

and so it explodes

Another look at the start of the Sacred Explosion

1 is unique

4 is cozy

108 is human

337,500 is personal finance

277,945,762,500 is global finance

79,301,169,838,123,235,887,500 is super-astronomical

a 38 digit number is getting transcendent

a 59 digit number has achieved a higher realm

108 is the human anchor of the Sacred Explosion

putting us up close and personal

with the vastness of numbers

the vastness of time and space

in which our numbers count and measure

Our courtesy-of-Python

window on numerical vastness

would cause Euclid to faint with surprise

Though he knew widely and understood deeply

geometry was aristocratic and sacred

while arithmetic was for shopkeepers

even Euclid could barely calculate

with the Greek number system

precursor to Roman numerals

which lacked zero and

computational algorithms

We experience the sacred number 108

through our practice

through its name

through its structure

through its place in greater realms

Our story of 108 concludes with a story

that starts with Euclid’s impotence

and ends with our power

Tragedy or triumph

with what mix of joy and sorrow

do we bow to the power of computation

that places 108 in the Sacred Explosion

that can reduce life to number?

Emptiness-inspired Indian mathematicians

recognized zero as genuine number

sunya in Sanskrit

cipher in Arabic

cero in Spanish

zero in French Italian Portuguese

created the system of numerals

we use to write “108”

developed algorithms for





Hail Brahmagupta

astronomer and mathematician

first to fully understood




His seventh century text

The Doctrine of Brahma

translated into Arabic a century later

brought the joy of calculation to Baghdad

We bow to Brahmagupta

and to those on whose shoulders he stood

The mathematicians of

the Baghdad House of Wisdom

improved and transmitted the art

of calculating with “Hindu” numerals

Hail al-Khwārizmī


whose ninth century text on arithmetic

survives only in Latin translation

made two centuries later in England

from whose name we derive




We bow to Muḥammad ibn Mūsā al-Khwārizmī

and to those on whose shoulders he stood

After Al-Khwarizmi appeared in Latin

a divide slowly formed across Europe

between algorists

who used his methods with Arabic numerals

to add subtract multiply divide

and abacists

who used Roman numerals

and employed devices with beads on wires

Enter Leonardo Bonacci “son of Bonacci”

Hail Fibonacci

widely traveled merchant of Pisa

shopkeeper and mathematician

first algorist known to history

whose 1202 text Liber Abaci

The Book of Computation

demonstrated the advantages of the

Hindu-Arabic system

in an array of “real life” problems

often involving money

showing the power of computation

to provide answers

to life’s persistent questions

remembered for a problem about rabbits breeding

the exponential population explosion

of the Fibonacci Sequence

with much tamer growth than

the Sacred Explosion

whose 8th term has 59 digits

while one has to go almost 300 terms

into the Fibonacci sequence

before attaining that magnitude

We bow to Leonardo Bonacci

and to those on whose shoulders he stood

Centuries after mathematicians and bankers

realized the advantages of Hindu-Arabic numerals

Roman numerals remained in use

108 was a notational device used to compute

CVIII was the real thing

but money talks and now

108 is the real number

while Roman numerals are used to give gravitas

to cornerstones and Super Bowls

We are reminded that in Europe

mathematics developed

in the service of money and war

while in Japan

Temple Geometry was cultivated for

intellectual artistic spiritual

pleasure and growth

Hail the sacred number 108

made holy and powerful by practice

made beautiful by symmetry

May its influence spread to

pacify enrich magnetize

and when appropriate destroy

all uses of number

We bow to 108


Newcomb Greenleaf

1:00 am

Saturday October 31, 1015

Saint Johnsbury, VT

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