Sunday, August 17, 2008

Milo Gardner - Personal Info

As the 11th and youngest child of Rea and Pansy Gardner early life was surrounded by siblings. Rea and Pansy were high school grads. Rea was born in Dec, 1891, and Pansy in May 1894. Rea graduated from BYU Academy, but poor financing ended a dream of graduating from BYU as happened to his father Sy. Rea was a sports enthusiast, checkers champ, historian, and religious philosopher (considering cosmology). Rea followed many of his father's interests (checkers champ, geography, the medieval era, the Silk Road, I Charlemagne, European Enlightenment and economics). Pansy, a musician (piano), motivated Don, Jack, and Allen (all clarinet), Colin (sax, clarinet), Faith (piano, song), Dixie (song, drums), and Zoe ( piano, song).

Born at 1364 Richins Avenue, Gridley, CA in 1938 with the aid of a mid-wife  Don and I arrived. Rotate the linked photo by clicking to see the home. Also move the 'stickman' to drive to Gridley 1/2 mile away. My ten siblings were delivered by my father, with MDs arrived to cut the cords. Our 7 boy, 4 girl family included Colleen, Colin's twin, died at birth. My mother, 1/16 native American, said her 11 home births arrived with little pain. Jack and myself arrived with no pain, per my mother's telling of the stories.

My early years were economically easier than my depression era siblings. Four brothers served in WW II. Don (Merchant Marines, Chief Boson's mate, Long Beach, Canal Zone, Florida, Norfork, Hawaii, and California coast); Hal (Pearl Harbor, welder); Jack (Sgt, Warrant Officer, and Captain Army Air Corp pilot flew new bombers to the Pacific); Vere (Navy gunner, by rifle downed a plane on 12/7 Pearl Harbor, and other planes in other  Pacific battles); and Allen (Marines, telephone linesman lived with Navajo Code Talkers on Okinawa,  restored telephone lines to Beijing in 1945-46).  Allen (Ag. teacher), Colin(Physics) and myself (Mathematics, MBA) are college grads. All seven sons may have gained college degrees except for WW II. Faith, Dixie and Zoe attended two years of college, and today too may have earned four year college degrees.

Reading came early, enjoying Prince Valiant and the comics before the first grade. School added westerns, local facts (i.e., Wilkes Expedition and Shinn's "Mining Camps") of pre-Gold Rush, and scholarly versions of Gold Rush people/events. Native Americans, and Wild West heroes and myths (ie Ishi, who walked into nearby Oroville in 1911, Wild Bill Hickok, and Kit Carson) books were read. From our backyard Indian grinding bowls and baskets were dug up. In the evenings we read articles from our family World Book Encyclopedia.

Movie going was enjoyed, often twice a week. Movies opened my eyes to the news of the world. Gridley's only theater was a main source of entertainment. I was often first in line. Once an older brother shook me hard when emulating Gentleman Jim Corbett's boxing skills (at age 5 or 6), showing little appreciation of fantasy as a motivation to set high goals in life.

As a farm boy serious work began at age ten. From that point I dreamed of another life, not enjoying dairy, custom hay cutting, drying, chopping, and blowing into barns, at 50% the cost of baled hay. My father also grew row crops and rice on his 90 acres, and 200 acres of rented land. Revenue and segmented costs per acre dominated discussions. Many hours were spent driving tractors, plowing, and discing, often all night, to avoid the heat of the day.

A social conscience developed early in labor and interpersonal areas. Gridley was the home of a county funded migrant labor camp. Weekly new "Okies" came to school, often with chips on their shoulders. To make friends playground/recess fights of some sort would take place. Winning the majority of those fights, many friends were made, almost never fighting with that person again. Administrators allowed the fighting, and sad "Okie" bashing that made my conscience cry.

During the summer swimming was big. Behind my home was an irrigation canal. The kids from south end of Gridley swam there seven days a week. There were other canal swimming holes in our area, places that the kids reached on bicycles. The town also had a swimming pool, and tennis courts.

Sports were an early passion. Grammar school championships in flag football, basketball and softball, were great fun. As an 8th grader, I broke a leg at half-time of a high school football game by playing tackle football. My head cheerleader sister (Dixie) was surprised to see an ambulance carry me off. High school was sports filled, entering at 5'tall and 100#. I Improved 880 and 1320 times though not near Vere's times (held a national class B record in the 1320). Baseball, and track were played two years, winning southern league 880 titles; and a JV football title another year. All year pick up games, mostly basketball and baseball, were played on local sports fields, usually at our middle school.
High school A, B, C, D basketball won C, and D titles, ended with a GIBT loss to El Cerrito (TOC champ) & Cornell Green, later of Dallas Cowboy fame.

Math was a special high school interest, as well as Spanish, chess and the sciences. History was fun, but my school only offered US history. I dreamed of traveling to Europe. College entrance exams returned a 50 percentile score. Being under financed, work before going to college was begun.

A 1956 summer AT &; SF railroad fireman job in hot Needles, Barstow CA, and Seligman, AZ, followed my mother's family trades. The job focused my attention on better working conditions. Work for work's sake was a bore, so a trip to a European dream via the US Army plan was put into effect.

Three years were spent in the US Army. Processed in, and out, at the Oakland Army Terminal. A meningitis outbreak at Fort Ord, ended a two week stay, riding a troop train to rainy Ft. Lewis, WA for basic training. We all worried about Quemoy, Matsu, and the revolt in Hungary, spots that we thought we would end up.

Advanced training was obtained at Ft. Devens, Mass, seeing pro sports for the first time, at Fenway Park (Ted Williams) and Boston Garden (Bill Russell, Bob Cousy). Qualifying for Monterey Presido's Army Language School, that would have brought me home for Christmas. Code breaking school was entered. Nosed out by two college grads, who stayed as instructors, cryptanalyst skills allowed Germany to be selected before any other classmate, based on my #3 class ranking.

Two years were spent in Bad Aibling, Bavaria, Germany. Three of us GIs co-owned a 1939 Opel, traveling to the French Riviera, Switzerland and Vienna. Our favorite city was Salzburg, 90 k away in neutral Austria, where uniforms were not allowed. We were civilians once or twice monthly. Train trips to Italy (Venice to Rome), Paris, and Holland, during tulip time were also enjoyed. Sports activities included company level basketball and fast pitch softball. Softball brought two championships, beating Special Forces and other nearby military units. I played 3rd and catcher.

A two month side trip to Lebanon (in the largest USA troop movement between the Korean War and Viet Nam), widened Russian language interests to include Arabic and cultures of the Middle East. Task Force 201 tent cities were set up in biblical olive orchards. Tents moved south of Beirut, near a beach. Daily swims were relaxing, as were chess tournaments. I was the chess champ. Time away from California would soon end.

After an Atlantic boat ride, seeing the Statue of Liberty, Ft. Dix, and throwing away a 10 cent pack of cigarettes overboard, never smoking again A three year enlistment happily ended 30 days early. An Aug. 1959 honorable discharge was awarded in time for the fall semester.

College entrance tests obtained 90 percentile results. The level motivated two college degrees, undergrad math degree (1964), and grad (1970). Ample electives allowed the history of economic thought, from the Ancient Near East and European definitions of commodities based monetary systems to pre-capitalistic trade, and modern economic systems were studied at CSU-Sacramento and CSU-Chico. Two sports were enjoyed in college. Our Chico State bowling team finished 10th at Oregon State in the western region finals. I averaged 180. Later a 190 average was held for a couple of years in a 25 year bowling hobby. Chess at the student union beat all students challenges ... losing only to my math diffy q. prof, #10 ranked in the USA a the time, became a grand master in his retirement was a  world class  chess player.

Post-college work began in aerospace, Vandenberg AFB and missile range issues, learning NASA mathematical astronomy. Bertha and I had dated four years. We married in 1965. Bertha is the second oldest of a 9 boy, 5 girl family. She was born in Durango, Mexico, moving to Gridley in 1955, the big flood year. Bertha and I enjoyed southern Cal for five years: beaches, Disneyland, Dodger, and Angels baseball games. Three years were spent working for Rockwell in Fullerton and Downey. For three years two evenings/week were spent at CSU-Fullerton.

Completing an MBA allowed a career change. We moved to northern California to be near our families. In northern California public health, the medical field, and public service were enjoyed. After 10 years my wife's family restaurant business called. As the CFO, a splitting of a seven brother chain into a three brother operation increased family associations.

Bertha and I are parents of three children. Tommy and Michelle arrived in Downey, and Anaheim. Missy arrived in Yuba City. Raising the kids was great fun. Tommy loved the trumpet, baseball, and fun. Michelle loved the clarinet, softball, basketball and fun. Missy loved softball, basketball, volleyball and quieter fun. Camping trips to the Sierras and softball to three national tourneys (2nd, 5th and 7th) were enjoyed. Missy, a center fielder at Cal Poly (SLO), was NCAA regional all-tourney, her last competitive day. Today Missy is a registered civil engineer. Michelle is a State of California tax collector. Tommy was in the Marines, having completed an electronic school in North Carolina, scheduled for an overseas assignment. In 1987 the day before leaving for Okinawa, Tommy died as a passenger in a traffic accident. Life is less fun these days. 

For several months following the accident time was spent in libraries studying unsolved math topics. A history of zero project upgraded a medieval topic that used zero as an exponent in base 10 decimals. Six months of readings pushed theoretical aspects of zero back 3,000 years to Egyptian engineering and accounting, Babylonian base 60 arithmetic systems, and Mayan arithmetic within a Chinese Remainder Theorem.

Chinese Remainder Theorems (CRT) aligned
Mayan calendars (unknown date), and China's calendar. A Chinese 'string of pearls myth' was confirmed by NASA, and modeled in Stellarium, rotating the sky to Feb. 2, 1950 BCE. China's CRT traveled the Silk Road to the Greek and medieval worlds, documented by Fibonacci in 1202 AD. Generally, world-wide numeration systems were read and enjoyed from Native Californians' 45 tribes (Kroeber's count), and North and Central American base four that stressed four cardinal directions in which Mesoamericans added base 5 to define base 20 systems that aligned growing season, moon, earth, several planets, and great cycle calendars within CRT applications (discussed by Floyd Lounsbury "The Sky in Mayan Literature" and others).

Code breaking has become a retirement hobby. Ancient math texts are decoded within cross-cultural situations. The first professional paper was published in 2002 on the
Egyptian Mathematical Leather Roll. Raw data from the 26 line text was translated into modern math/arithmetic in ways that went beyond additive  Eurocentric limitations documented in Crest of the Peacock. Additive 20th century scholars applied transliteration rules that garbled the EMLR and RMP raw data. Occam's Razor's assist in ungarbling the data. 

For example, several unsolved 20th century Egyptian math problems have been solved. One was the ancient arithmetic foundation of 2/n tables. The problem was parsed by small ad hoc internet teams. Team members created meta and micro statements that translated into modern arithmetic statements validated by ancient texts. Meta and micro formulas linked two, three and four ancient texts in ways that are inconsistent with 1920s additive Eurocentric scholars

Occam's Razor has assisted code breaking projects by sorting fragmented meta and micro arithmetic statements into readable statements, tested against ancient texts from the same time period. The razor points out the simplest ancient methods. Meta statements summarized readable arithmetic, algebra, geometry, arithmetic progression and weights and measure statements. Micro statements reported unclear and readable fragments of cubit units, pesu units, and so forth.

In 2004 and 2006 Occam's Razor, and trivalent logic of languages (i.e. Aymara) defined a new analysis method. The method was applied to the Egyptian math problem. Confirming the decoding of hard-to-read problems and embedded arithmetic motivated a professional paper on the Akhmim Wooden Tablet (AWT). Trivalent logic opened a cross-cultural decoding door that places observers, decoders and translators outside of the hieratic written language. In addition ciphered hieratic mathematical data were seen as meta forests, pointed out three classes of quotient (Q) and scaled remainders (R) method in the AWT, the RMP and related hieratic math texts.

A 2006 paper parsed five 1900 BCE AWT division problems that included five proofs. A two week analysis of a raw arithmetic structure of divisions of hekat volume units began and ended with  (64/64), a hekat unity. Conclusions were confirmed by 40 data points in the 1650 BCE Rhind Mathematical Papyrus(RMP), and five data points in the AWT. Scribal multiplication and division operations were dominated by two binary quotients (Q) and scaled (5/5) remainder (R) statements, (64/64)/n and (6400)/64)/n reported by  Q/64 + (5R/n)ro two-part answers, 29 times in RMP 81 (column 1). A simple Q + R/n quotient and remainder was reported when hin, 10/n,  and ro, 320/n,  data was divided in the RMP, 29 times in RMP 81 (column 2).

Additional corrective studies include theoretical views of Egyptian arithmetic that scribes used to scale grain inventories. Egyptology scholars consulted included the 2002 work of Hana Vymazalova and Tanja Pemmerening. Working an informal interdisciplinary teams have added to my personal skills, as well as parsing the larger ancient Egyptian mathematical topic.

By 2009 it became clear by working with Tanja Pemmerening, and others, that Middle Kingdom remainder arithmetic 'healed' a binary round-off problem. In terms of a volume unit, a hekat was written as (64/64), a unity, a 2002 point stressed by Hana Vymazalova, and divided by 3, 7, 10, 11, and 13, within the formula:

(64/64)/n = Q/64 + (5R/n)*1/1320, and 100-hekat by (6400/64)/n

In the AWT, n= 13 was written as 64/13 finding a quotient( Q) as 4, and a remainder (R) as 12, such that:

(64/64)/13 = 4/64 + (60/13)*(1/320)

= 1/4 + (4 + 1/2 + 1/13 + 1/26)*1/320

In other 2009 studies, RMP 35 through RMP 38 data  parsed 1650 BCE scribal thinking by defining one hekat as 320 ro. The 320 ro unity was by n by writing:

320/n ro = (Q + R/n)*ro

and by multiple hekat volumes by

320m/n = Q + R/n

For example, 10 hekats of fat, 3200 ro, was divided by 365. A daily usage rate was found in hekats per:

8 + 280/365 = 8 + 2/3 + 1/10 + 1/2190

The quotient 8 was proven by the scribe by writing: 1-365, 2 - 730, 4- 1460, 8 = 1920. The numerator of the remainder, 3200 - 2920, was proven by 243 1/3 + 365 1/2 + 1/6 = 280, since 2920 + 280 = 3200.

as the AWT defined a hekat division as (64/64)/n, and in RMP 47 (6400/64)/n. Unities of various types were important to Ahmes.

Two theoretical aspects of  unit fraction arithmetic, covering 2850 years, were confirmed from 2005 to 2009 by parsing RMP 36, RMP 37 and the Liber Abaci.  Ahmes in 1650 BCE recorded a  RMP 2/n table that contained 51 conversions of rational number 2/n to concise unit fraction series. Ahmes associated each 2/n with a least common multiple (LCM) m to obtain 2m/mn. Numerator 2m was written as the sum divisors of denominator mn before a concise unit fraction series was recorded. In 2005 the 1202 AD Liber Abaci (LA) was parsed in terms of the 2,850 year older 2/n table method. The LA described closely related LCM m in a subtraction context converting rational number n/p by (n/p - 1/m) = (mn -p)/mp.

In 2009 a study reviewed a 1650 BCE Egyptian text that parsed 87 Rhind Mathematical Papyrus problems. Egyptian arithmetic contained two-sides, one theoretical, and the other practical. This study added back overlooked theoretical aspects, as well as correcting muddled practical statements. 20th century additive statements had been muddled by scholars that published transliterations of math texts by only reporting practical considerations. Several 20th century errors have been corrected by pointing out attested arithmetic operations used to scale rational number answers reported in arithmetic, algebra, geometry, arithmetic progressions, weights and measures, and pesu (proportional) problems.

In 2010, a scribal pesu was decoded in RMP 69-78. The same proportional elements were decoded in the Berlin Papyrus to solve two second degree equations. The common content of RMP 69 and the Berlin Papyrus were published by H. Schack-Schackenberg in 1900.  20th century scholars have read Schack-Schackenberg's decoding method. For personalized reasons several scholars do not accept it. The historical role of the pesu's proportional relationship to four  MMP 10, RMP 41,42, 43  and Kahun Papyrus  algebraic geometry formulas adds validity to Schack-Schackenberg's approach.

Three professional papers have been published on the theoretical side of Egyptian mathematics. Two papers describe 1850 BCE Egyptian Mathematical Leather Roll (EMLR), and 1900 BCE AWT theoretical methods. Decoding papers were published in India in 2002, and 2006. A condensed EMLR paper was published in Germany by Springer: Encyclopedia of the Non Western History of Science, and Medicine, 2005. An AWT paper was published by Ganita Bharati, Bulletin of the Indian Society for the History of Mathematics, Vol 28 (2006).

Online publications consist of:

A. Egypt

1. Why Study Egyptian Mathematics?  (Planetmath)

i.e. Ahmes Papyrus(blog); (Planetmath)
  ; (Wikipedia); (Mathworld)

2. Akhmim Wooden Tablet (blog), Planetmath

3. Ahmes Bird-feeding rate problem (Planetmath)

4. Berlin Papyrus (Planetmath), (Wikipedia)

5. Economic Context of Egyptian Fractions (Planetmath)

6. Egyptian Mathematical Leather Roll (blog); (Wikipedia); (Mathworld); (PlanetMath)

7. Egyptian fractions (Planetmath), History of Egyptian fractions (blog), Vulgar fractions (blog)

8. Egyptian Fractions, Hultsch-Bruins Method (Planetmath)

9. Egyptian Geometry (Planetmath)

10. Egyptian multiplication and division, Wikipedia

11. Heqanakh Papyri (Wikipedia)

12. Hekat Division (Planetmath)

13. Kahun Papyrus (Wikipedia),(Planetmath)

14. Least Common Multiples, modern and ancient (blog)

15. Red Auxiliary Numbers (Wikipedia), (Planetmath)

16. Reisner Papyri (blog); (Wikipedia)

17. RMP 2/n table, (blog) Wikipedia, RMP 36 and the 2/n table, Planetmath, blog

18. RMP 35-38 and RMP 66, RMP 36 (Open University forum)

19. RMP 53-55 (Planetmath)

20. RMP 69 and the Berlin Papyrus

21. Remainder Arithmetic (Planetmath)

22. Remainder Arithmetic vs Egyptian Fractions (Planetmath)

23. Webinar,  a one hour video response to the Feb. 2010 BBC program, was recorded on July, 2010 was offered to the math education community.

24. Oldest Puzzle, per the New York Times

B. Classical Greek

25. Plato's Mathematics (Planetmath), Math Forum

26. Archimedes' Calculus (Planetmath)

27. Hibeh Papyrus (Planetmath)

C. Medieval

28. Arabic Numerals (Planetmath)

29. Liber Abaci (blog); (Planetmath)

C. Mesoamerica

30. Aztec Surveying (blog)

31. Mayan Super Number Arithmetic  (Planetmath)

32. Mayan Seasonal Almanc (Planetmath)

D. Other (Astronomy)

33. Acano Lunar Calendar Method (Planetmath)

D. Wikipedia Bio  and WW II deeds of four my brothers:  Four Stars in the Gardner Window

Our family has grown to five grand children. Michelle and Tony have three boys, Missy and Scott are raising two girls. Chris (27) has a full time job with Golden One Credit Un
ion and is a part-time college student in business.  Erik (25) was honorable mention all-Metro linebacker. In college Erik played Div II rugby for Sierra College. The Wolverines finished 4th in the 2011 . The finals were in  Pittsburgh, PA. I  watched it all on streaming TV. Erik installs windows in new and upscale homes, a skill he picked from has father, a contractor.

Adam is a freshman at Del Oro in Loomis,  plays football on an undefeated team and wrestles at 195 pounds, also reached the masters level as Del Oro won the 2016 section masters title. Avery is a 8th grader and Hannah is 6th grader in Orangevale. In soccer Avery and Hannah do well in competitive leagues, on the premier level.  Both girls play for Ethos, in El Dorado Hills. Scott coaches Avery's team.  Family and sports fill the grand parents' time leaving plenty of time for hobbies, and family vacations.


Th. said...


Well met.

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SasQ said...

How to contact you? Do you have any e-mail address?

queridia said...

Hello Milo. Isn't the Internet wonderful? It's obvious you've had a very adventurous life, overall a happy family life and an ongoing interest in ancient thought. The blog is very nice.

SasQ said...

Hello. I'd like to contact you, because I'm searching for any proofs that ancient Egyptians really used the Eye of Horus symbol elements as unit binary fractions. I cannot find any proof of that, nor confirmations of it in historical sources (papyri). Can you give me any proofs? And do you have some good photos of these papyri you mention? Because I cannot find any good quality (that is, readable) images of them.
I've sent you also a message at with my e-mail address.

Marcela said...

Hi Milo. I am just reading about The Rhind Mathematical Papyrus and I saw your contribution on line on - History of the world in 100 objects -
It does not seem to me that BBC came back to you. Am I right?
Thank you.