Summary with PHI

'In the course of normal science, it may happen that anomalies begin to accumulate. Some of these may be set aside for future research. Some may be dismissed as irrelevant. But if a sufficient number of anomalies accumulate, anomalies, which resist solution by the paradigm or incorporation into it, a crisis develops. As the crisis intensifies, scientists begin to offer and promote new paradigms capable of accommodating the anomalies. If one of these paradigms attracts the attention of a sufficient number of members of the research community, a scientific revolution takes place. The research community learns to see things in a different way. It develops a new set of methods and concerns. Kuhn points out that unless there is a recognizable crisis, provoked by an accumulation of crucial anomalies, there will be no movement to a new paradigm. The first step toward movement to a new paradigm is thus recognition of anomalies, of counterinstances to the current paradigm' (Michael Cremo).

In the United States, as elsewhere, mainstream concepts of history are under assault and there is a growing awareness that the accumulation of archaeological anomalies is forcing a major paradigm shift. Primary resistance to the shift is spearheaded moreso by the academic community than the by the indigenous population, who have attempted for years to tell the mainstream scholars of "other historical factors".

North American Indian historian, Vine Deloria jr. is righteously indignant about the hijacking of history by do gooder paternalist Europeans, who are creating a cozy, socially engineered pseudo-history for the Indian tribes of North America.
He sees this as European led condescension toward his people and complains, 'There's no effort to ask the tribes what they remember of things that happened.
He goes on to say, 'numerous tribes do say that strange people doing this or that came through our land, visited us, and so on. Or they remember that we came across the Atlantic as refugees from some struggle, then came down the St. Lawrence River and so forth. There's a great reluctance among archaeologists and anthropologists to break centuries-old tradition and to take a look at something new...As for the history of this hemisphere from say, five thousand B.C. forward to our time, the mainstream scholars just don't want to deal with that at all. Let me give you an example. Years ago I spoke at an academic archaeological conference, and at the end of my speech I asked, 'Why don't you guys just drop the blinders and get into this diffusionist stuff?' My host, David Hurst Thomas, just about lost it and said, 'Do you know how long and hard we've fought to get members of this profession to admit that Indians could have done some of these things? And now you're saying it was Europeans!'''.

A similar set of limiting parameters has been foisted onto us in our "New Zealand" interpretation of regional history. Again, social engineers have oftimes discounted as irrelevant and unreliable the oral traditions of the Maori Elders and have taken a paternalist approach to creating pseudo-histories that promote limited, santitised versions of Maori history, to the exclusion of all else that went before.
All over the New Zealand landscape we are finding standing stone structures used for astronomy, navigation and land-mapping. There is evidence of "Beehive house villages and a large, former "Stonebuilder" population

Local historical interpretation is locked into a mere 800-900 years of human occupation, with a marked reluctance to accept any significant occupation at earlier epochs...in the face of, what many amateur researchers consider to be, dynamic evidence of at least 5000 years of continuous human presence in New Zealand.
Regional, social engineering of history is as much a condescending insult to the New Zealand Maori people as it is to Deloria's North American Indian tribes-people, in that it discounts their oral traditions and histories as being of little worth or substance. We seem to be enveloped by an academic attitude of "forced amnesia", which makes us forget truths that were general knowledge and openly discussed up until about 3 decades ago.

New Zealand historical resources carry many recorded references related to a large population of pre-Polynesian "red headed, light complexioned Tangata Whenua", albeit mostly in rare, out of vogue books buried in the backrooms or basements of our libraries and archives. One can also source such references, if one asks politely, from the Maori Elders who have memorized the oral traditions of their Iwis (Maori tribal groups).

THE MAORI WHAREWAENANGA (SCHOOL OF LEARNING).

It is the purpose of this study to show evidence of an early pre-Polynesian people, resident in New Zealand, whose former presence is attested to by many ancient ruined structures from the pre-Maori epochs. This evidence lies scattered across the length and breadth of New Zealand. It can be demonstrated that aspects of a regionally identifiable, Northern Hemisphere knowledge was handed down, at least in part, and survived within the Maori Wharewaenanga.

The dilapidated Crosshouse at Miringa Te Kakara, central North Island, New Zealand, shortly before it was burned down in 1983. Photos appearing in this article courtesy of Jan & Ron Raison.

Let us turn our attention to one such Wharewaenanga, School of Learning, established by the Pao-Mirere or Hau Hau movement at Miringa Te Kakara in the center of the North Island.
There appears to be some confusion as to the date of this building's construction and disagreement amongst the Maori Tohungas (sages) as to whether or not Pao-Mirere activity, related to the "Crosshouse", was to construct an entirely new building or simply to renovate to an older existing structure.
It is known that a series of Wharewaenanga structures bearing the name Miringa Te Kakara or Waerenga A Kakara had existed for centuries. Similar "temple" structures were located at 3 other regional centers contempory with the era of the Crosshouse.
The building, initiated by King Tawhiao's directive to Chieftainess Ngaharakeke, is said to have been completed in about 1865. Bishop Thomas Herangi, guardian of the Crosshouse up until the 1980's, cited evidence of the star temple having been built in 1682, with renovations occurring in 1788 & 1887.

The main interest in this building must lie in its geometric and measurement attributes, which code astronomical and navigational knowledge found from Egypt to Great Britain and North America.
The key to extracting the international parcel of codes lies in an understanding of the common measurement standard being used internationally by an ancient, highly mobilised migrating group, who set up colonies all over the globe. Many of these colonies appear to have begun as metal ore mining operations.

The Crosshouse of Miringa Te Kakara was not some form of "orphan" dwelling, the attributes of which are unique to New Zealand. It was built according to an internationally distributed parcel of astronomical codes, in accordance with a measurement standard that migrated throughout the ancient world and spanned oceans.

Careful, in depth surveying research into the dimensions of astronomical sites over several continents shows that the so-called British Standard of measurement isn't originally British at all, but is Egyptian/ Sumerian/ Babylonian. This concept can be severely and finely tested mathematically by trigonometry on internationally distributed structures built to this singular measurement standard… from the Giza Plateau to Stonehenge to the Octagon of Newark Ohio… to the Crosshouse at Miringa Te Kakara.

THE DIMENSIONS OF THE CROSSHOUSE.

Over the years several publications have mentioned close approximation measurements of the Crosshouse, like the 1959 Journal of the Polynesian Society article, which gives the following description.
'This building is 54 ft in length along the north-south and east west axes. Each wing is 17 feet wide and the height to the ridge poles is 11 ft. 6in.
Each wing has a small open sided porch and entrance to each is gained through a sliding door hewn from one piece of timber 4 ft high, 2 ft wide and 2 ins. thick...Each portal is fitted with two ports to admit light and these are closed by solid wooden slabs about 18 ins. square and 2 ins. thick, which are slid into position....The walls...are about 6 ft. high. Inside the building will be seen 5 uprights, which have been set in the ground to support the ridge poles...the centre pole is about 6 ft. in circumference while the other 4 are 18 ins. in circumference. At the inner corners of the cruciform structure are some L-shaped corner pieces 2 ins. thick... About 6 ft. in from each wall are footboards about 12 ins. wide and 1½ ins thick.

The most accurate, detailed plan ever made appears to be that of architect C. G. Hunt in 1958 for his 1959 article in the Journal of the Polynesian Society, 68: 3-7. Hunt's scaled plan has been used in this mathematical analysis of the Crosshouse. The full analysis tests geometry detectible at the Waitapu standing stone circle in Northland, NZ, against similar geometry found at Rennes le Chateau in Southern France by former army surveyor David Wood and his team. Careful work undertaken in the Languedoc Province showed very clear trigonometric evidence of known British Standard measurements like the mile. David Wood's work verified that a Celtic cubit of exactly18 British Standard inches existed in the placement of ancient structures of the Renne's valley and the measurement standard found there was the modern British one.
It can now be demonstrated that "slightly drifted" "weights, measures & volumes" of the foremost civilizations of the Mediterranean have stemmed from one common source in Egypt and the British system is the most intact survivor of the original universal system. Even the strange Egyptian cubits are fully a part of the "so-called" British Standard of Measurement.

The Plan of C. G. Hunt was for the perceived purpose of "restoration" of the Crosshouse building and did not take into account the structure's correct orientation within the greater landscape. Hunt's plan shows the building's wings orientating onto North, South, East & West, whereas 2 wings actually lay along the observable rise/ set line of Winter solstice sunrise and Summer solstice sunset at 60-degrees and 240-degrees respectively. The other two wings lie, therefore, at azimuths of 150-degrees & 330-degrees.

In consideration of the slightly elevated hills to the NE & SW, it's probable that the Winter solstice sunrise and the Summer solstice sunset were observed from the center pole or secondary poles through the open doorways. It's also highly likely that the northern-most Lunar Standstill rise and the southern-most Lunar Standstill set could be observed through the elevated windows, with the observer seated to the front or side of the center pole.

The exact position of the Crosshouse's center pole is determined by the distant peak of Mt. Ranginui, which lies directly north and provides the benchmark fix for the structure and its outlying nui poles. The Crosshouse was described as a place for "conducting the rites as in old times".
The mathematical attributes of the Crosshouse show us very clearly what many of those rites were and how they parallel astronomical and navigational knowledge preservation activities undertaken by priests within the great civilizations of the Northern Hemisphere. The knowledge coded into the Crosshouse at Miringa Te Kakara has a direct pedigree back to Egypt, Great Britain and North America, as elsewhere.

The correct orientation of the Crosshouse, using the peak of Mt. Ranginui as the benchmark for true north from the building's center pole. Two wings were set out along the line of the Winter solstice sunrise and the Summer solstice sunset positions, which, in consideration of slightly elevated outer terrain, should have been observable at 60-degrees and 240-degrees respectively.
It also appears apparent that an observer seated at the base of centre pole, viewing upward at an angle of about 7.5-degrees through the NE window would observe the Major Northern Standstill position of moonrise and through the SW window the Major Southern Standstill moonset. Alternatively, the observer could sit on seats to the right and left of the pole and watch not only the major standstill rise and set but, plausibly the minor standstill rise and set also through the adjacent window on the other side of the doorway.
This concept is supported by a late era tradition of reflecting direct moonlight beams from paua shells onto incised symbols within the building. All indicators suggest that the 18.613-year lunar cycle was being tracked.

Drawing by W.A. Taylor of how the cruciform building looked when in pristine condition.

GEOMETRIC PRINCIPLES.

The 1958 Crosshouse survey plan of C. G. Hunt seems to indicate a slight elongation of two wings, which allows them to sit within a circle of 56 feet and the 90-degrees opposed wings to sit within a circle of 55 feet. These numbers are of extreme importance to navigation or calendar counting systems found from Egypt to Great Britain and North America. Let's now consider some mathematical attributes:

PHI reducing circles and squares define significant positions within the Crosshouse.
From Egypt's Great Pyramid to Stonehenge in Southern England the geometric principle of PHI reducing circles and squares has been observed to be a significant factor in the defining geometry of observatories or code bearing structures.

PHI relationships at Stonehenge… Avenue marker, Aubrey Circle, "Y" Holes circle, Sarsen Circle, etc.

The ratio between the upper surface area of the Great Pyramid (assuming a symbolic capstone) to the surface area that the Great Pyramid occupies at its base = 1.6180339 (PHI) to 1. The PHI method for arriving at a side diagonal value of 611.617 feet is: 378 feet (½ the base length) X 1.6180339 (PHI).

SIGNIFICANT DEGREE ANGLE CODES WITHIN THE CROSSHOUSE.

It is known that the Sumerians, Babylonians, Greeks and other great civilizations of antiquity, used an ancient system of dividing a circle into 360-degrees or 720 calibrations. Their method persists in modern times. The Crosshouse at Miringa Te Kakara (Tiroa) incorporates the same system and clearly encrypts solar, lunar and navigational codes (via the degree angle numbers generated) used universally in the ancient Northern Hemisphere.

A very important code found from Egypt to Great Britain North America and South America is based upon 11.52. Under the ancient Babylonian system of time measurement (days, hours, minutes and seconds) the Heliacal rise of Sirius gave a close approximation of the Solar year @ 365.25-days. From this figure was deducted 11.52 minutes to render the figure 365.2420-days (Mayan Calendar). The red vectors are expressions of 11.52 increasing at intervals of 90-degrees. The second increase in the series designates the azimuth angle to Puriora Mountain, indicating that the Crosshouse was precisely positioned to be (1) due south of Mt. Ranginui and (2) 11.52 + 270-degrees WNW of Puriora Mountain's peak.

Each of the blue vector azimuths produce extremely important astronomical/ navigational numerical codes used over several continents in antiquity. An example of this (288) relates to The Great Pyramid, Menkaure Pyramid, Stonehenge's Aubrey Circle and astronomical codes encrypted into the Bible in Num. Chpt. 7. Code encryption appears to have been the true preoccupation of many biblical writers.
The degree angle spread between the two edges of these very prominent, large wrap around corner moldings was obviously intended to convey 6.48-degrees, using the center post as the fulcrum. The 6.48 code is of extreme importance within an ancient mathematical progression, which relates to the duration of the Precession of the Equinoxes…6.48, 12.96 (½ the precessional cycle was 12960-years), 19.44, 25.92 (Precession was calibrated to endure for 25920-years). Another important number associated with this progression includes 51.84 (angle of the Great Pyramid's sides & 1/500th of the duration of Precession in years). It is important to note that the 6.48-degree increment features strongly at the huge Octagon earthworks geometric complex at Newark Ohio and features within the chosen design angles of 8 embankments. One of the embankments resides on an azimuth angle of 64.8-degrees.

A part of a "wrap around" corner molding can be seen to the right hand side of the above picture. It will be noticed that half of a strange geometric design is etched into the timber, with the rest of the design continuing around the corner onto the 90-degrees opposed face. Each large corner molding, as well as the centre post, had either the same (4-pointed star) geometric design carved into it or a variation of the design.

Degree angle codes as they relate to the sides of the windows. All of the numbers shown are highly important to ancient systems of calendar counting, navigation and determining the size of the Earth… or counting days within the 18.613-year lunar nutation cycle.
The occurrences of degree angles related to 50.4, 230.4 and 340.2 are especially dynamic, as these are amongst the most important codes of antiquity and are found upon the Great Pyramid, Khafre Pyramid, at Stonehenge or within the Octagon of Newark, Ohio.

STAR & CROSS PATTERN GEOMETRY.

The work of David Wood and his team in Rennes Le Chateau in the Languedoc Province of Southern France, showed that overland positions of ancient structures situated within the Rennes valley, demonstrated star and cross pattern geometry. A huge hexagram, pentagram, an 8-pointed star and several forms of crosses were clearly identifiable in the carefully surveyed former positions of ancient "Pagan" sites located there.

What David Wood identified is repeated in whole or in part at individual observatories of coded structures around the globe. Many aspects of his team's findings are most certainly incorporated into the Crosshouse at Miringa Te Kakara and represent a standard astronomical system taught to initiates. Knowledge of the universal system allowed accurate astronomical/ navigational calculators to be built worldwide.
The ancient people were very practical and every star or cross pattern had a prescribed, useful purpose that would aid in the preservation of knowledge and optimization of regional functioning society.

THE 8-POINTED STAR.

During a survey of 1/7/2001 it was discovered that there were 4 additional posts close to the internal corners of each wing, which surveyor C.G. Hunt had not included on his 1958 plan. These were found to reside just over 14 feet from the center in the positions shown. Also the secondary posts shown on Hunt's plan were measured to be a few inches closer in than where Hunt had marked them. This infers "expert consultation" between Hunt and Pao-Marire Kaumatua's as to the correct positioning of those posts.

The 8-pointed star in normal North-South orientation would have its rays extending toward the primary and secondary positions of the compass. In the geometry of the Crosshouse the rays extend to 15, 60 (solstice), 105 (reed), 150, 195 (calendar), 240 (solstice), 285 and 330 (navigation) degrees respectively.

Note how the 8-pointed star geometry relates to positions within the building. Returning lines intersect with the edges and centre of the 4 corner moldings and the 3rd inward PHI circle is perfectly contained within the star. The ends of the walkways terminate widthwise onto the ray lines of the star.

In Egyptian mythology the 8-pointed star is associated with Isis…also known in the region as Ishtar or Astarte.

THE GRAND CROSS.

Amidst the geometry detected at Rennes Le Chateau by David Wood and his team was a Grand Cross.
This is also clearly in evidence at Miringa Te Kakara, where the width of each cross arm, designated by the extreme positions of the Crosshouse windows, is 24-degrees of arc.
There are 8 intersection positions on the 8-pointed star that clearly stipulate how the Grand Cross is constructed and how it overlays the 8-pointed star geometry.
This cross remained prominent in the symbolism of many ancient Northern Hemisphere civilizations and developed into the Knight's Cross, the Templar Cross, etc.
This cross, when combined with an overlaying PHI circle is recognizable as the Celtic Cross.

The Grand Cross, clearly designated by extreme window positions within the Crosshouse. Overlaying PHI circles, adding to the geometry, create the Celtic Cross…much displayed in regions like Ireland.

The Grand Cross overlaying the 8-pointed star. Note how intersections of the star lock the cross pattern firmly into position. The diagonal vectors numerical values within this cross pattern are highly important to known ancient astronomical/ navigational methodologies.

THE 6-POINTED STAR & THE 12-POINTED STAR.

These two star patterns were highly important to tracking the sun at the solstice or fixing onto the constellations of the zodiac (12 houses…30-degrees apart).

This is a very useful star pattern, especially at latitudes like Miringa Te Kakara, inasmuch as the ray points of the star fix onto the 4 rise and set positions of the solstice sun, with the 2 additional points designating north and south. Near Eastern tradition seems to suggest that this was the star pattern assigned to RA the Sun God, which would make perfect sense.

The 6-pointed star doubles out into the 12-pointed star, thus designating the positions of the 12 houses of the zodiac, with a spread of 30-degrees of arc for each house. Note how intersections of this star comply with the internal wall line of the Crosshouse.

THE CROSS OF SET.

David Wood and his team detected a thin cross within the marked positions of Rennes Le Chateau. A central PHI circle and intersections within the layered geometry of the Crosshouse attest to the presence of this thin cross in the geometric design of the Crosshouse. Whereas the Grand Cross has an arm thickness of 24-degrees, the thin cross extends through 12-degrees.

THE "ALL IN" GEOMETRY SO FAR.

The Maori religious movement "Ratana", which has a strong affiliation to the Pao-Marire movement of the 1860's, continues to use symbols or patterns borne out of the geometry of the Crosshouse as religious emblems of their faith. These symbols were handed down from the ancient Patu-pai-arehe, some of who were ancestors to the Maori descendants from the Takitimu canoe. The main group of Patu-pai-arehe in residence at Tiroa was finally vanquished from the valley region into the high seclusion of the Rangitoto Ranges.

It becomes evident as to why 4 special post positions were made near the outer corners where each wing began. In the above picture the posts (red) are shown to be major intersections for crossing geometry. This star and cross pattern geometry is steeped in antiquity and both the squatting god (Ammon RA) of the Egyptian Hypocephalus funerary amulet or the squatting god of the New Zealand Moriori people stem from a design created by parts of this geometry. How this occurs is explained in detail within this website (see Article, Megalithic New Zealand parts 1-6).

CREATING PHI GEOMETRICALLY AT THE CROSSHOUSE.

The windows of the Crosshouse provide the necessary 18-degrees of spread to easily create 2 separate sets of PHI relationships between a circles and squares with the diameter of the square ½ PHI less than that of the circle.
Ropes strung between the internal corners of the porches gave a reading to create an accurate PHI overlay for the 55 feet circle (blue) and two wings were a ½ PHI reduction to a 55 feet circle. Two additional wings related to an encompassing circle of 56 feet (red) and a square reduced by ½ PHI of that diameter designated the internal length of the building for the wings concerned.

CONTINUE

Fibonacci Numbers and Nature

This page has been split into TWO PARTS.

This, the first, looks at the Fibonacci numbers and why they appear in various "family trees" and patterns of spirals of leaves and seeds.

The second page then examines why the golden section is used by nature in some detail, including animations of growing plants.

Contents of this Page

The icon means there is a Things to do investigation at the end of the section.

• Rabbits, Cows and Bees Family Trees
  • Fibonacci's Rabbits..
  • Dudeney's Cows
  • Honeybees and Family Trees
• Fibonacci Numbers and the Golden Number
• Fibonacci Rectangles and Shell Spirals
• Fibonacci numbers, the Golden Section and plants
  • Petals on flowers
  • Seed heads
  • Pine cones
  • Leaf arrangements
    • Leaves per turn
    • Leaf arrangements of some common plants
  • Vegetables and Fruit
• Fibonacci Fingers?
• A quote from Coxeter on Phyllotaxis
• Navigating through this site
• References and Links

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

Rabbits, Cows and Bees Family Trees

Let's look first at the Rabbit Puzzle that Fibonacci wrote about and then at two adaptations of it to make it more realistic. This introduces you to the Fibonacci Number series and the simple definition of the whole never-ending series.

Fibonacci's Rabbits

The original problem that Fibonacci investigated (in the year 1202) was about how fast rabbits could breed in ideal circumstances.

Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was...

How many pairs will there be in one year?

1. At the end of the first month, they mate, but there is still one only 1 pair.
2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
4. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.

Can you see how the series is formed and how it continues? If not, look at the answer!

The first 300 Fibonacci numbers are here and some questions for you to answer.

Now can you see why this is the answer to our Rabbits problem? If not, here's why.
Another view of the Rabbit's Family Tree:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

The Rabbits problem is not very realistic, is it?

It seems to imply that brother and sisters mate, which, genetically, leads to problems. We can get round this by saying that the female of each pair mates with any male and produces another pair.
Another problem which again is not true to life, is that each birth is of exactly two rabbits, one male and one female.

Dudeney's Cows

The English puzzlist, Henry E Dudeney (1857 - 1930, pronounced Dude-knee) wrote several excellent books of puzzles (see after this section). In one of them he adapts Fibonacci's Rabbits to cows, making the problem more realistic in the way we observed above. He gets round the problems by noticing that really, it is only the females that are interesting - er - I mean the number of females!

He changes months into years and rabbits into bulls (male) and cows (females) in problem 175 in his book 536 puzzles and Curious Problems (1967, Souvenir press):

If a cow produces its first she-calf at age two years and after that produces another single she-calf every year, how many she-calves are there after 12 years, assuming none die?

This is a better simplification of the problem and quite realistic now.

But Fibonacci does what mathematicians often do at first, simplify the problem and see what happens - and the series bearing his name does have lots of other interesting and practical applications as we see later.
So let's look at another real-life situation that is exactly modelled by Fibonacci's series - honeybees.

Puzzle books by Henry E Dudeney

Amusements in Mathematics, Dover Press, 1958, 250 pages.
Still in print thanks to Dover in a very sturdy paperback format at an incredibly inexpensive price. This is a wonderful collection that I find I often dip into. There are arithmetic puzzles, geometric puzzles, chessboard [uzzles, an excellent chapter on all kinds of mazes and solving them, magic squares, river crossing puzzles, and more, all with full soutions and often extra notes! Highly recommended!

536 Puzzles and Curious Problems is now out of print, but you may be able to pick up a second hand version by clicking on this link. It is another collection like Amusements in Mathematics (above) but containing different puzzles arranged in sections: Arithmetical and Algebraic puzzles, Geometrical puzzles, Combinatorial and Topological puzzles, Game puzzles, Domino puzzles, match puzzles and "unclassified" puzzles. Full solutions and index. A real treasure.

The Canterbury Puzzles, Dover 2002, 256 pages. More puzzles (not in the previous books) the first section with some characters from Chaucer's Canterbury Tales and other sections on the Monks of Riddlewell, the squire's Christmas party, the Professors puzzles and so on and all with full solutions of course!

Honeybees and Family trees

There are over 30,000 species of bees and in most of them the bees live solitary lives. The one most of us know best is the honeybee and it, unusually, lives in a colony called a hive and they have an unusual Family Tree. In fact, there are many unusual features of honeybees and in this section we will show how the Fibonacci numbers count a honeybee's ancestors (in this section a "bee" will mean a "honeybee").
First, some unusual facts about honeybees such as: not all of them have two parents!
In a colony of honeybees there is one special female called the queen.
There are many worker bees who are female too but unlike the queen bee, they produce no eggs.
There are some drone bees who are male and do no work.
Males are produced by the queen's unfertilized eggs, so male bees only have a mother but no father!
All the females are produced when the queen has mated with a male and so have two parents. Females usually end up as worker bees but some are fed with a special substance called royal jelly which makes them grow into queens ready to go off to start a new colony when the bees form a swarm and leave their home (a hive) in search of a place to build a new nest.

So female bees have 2 parents, a male and a female whereas male bees have just one parent, a female.

Here we follow the convention of Family Trees that parents appear above their children, so the latest generations are at the bottom and the higher up we go, the older people are. Such trees show all the ancestors (predecessors, forebears, antecedents) of the person at the bottom of the diagram. We would get quite a different tree if we listed all the descendants (progeny, offspring) of a person as we did in the rabbit problem, where we showed all the descendants of the original pair.

Let's look at the family tree of a male drone bee.

1. He had 1 parent, a female.
2. He has 2 grand-parents, since his mother had two parents, a male and a female.
3. He has 3 great-grand-parents: his grand-mother had two parents but his grand-father had only one.
4. How many great-great-grand parents did he have?

Again we see the Fibonacci numbers :

                                       great-     great,great   gt,gt,gt
                          grand-      grand-     grand         grand
Number of       parents:   parents:    parents:   parents:      parents:
of a MALE bee:    1           2           3          5             8
of a FEMALE bee:  2           3           5          8            13
 

The Fibonacci Sequence as it appears in Nature by S.L.Basin in Fibonacci Quarterly, vol 1 (1963), pages 53 - 57.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

Fibonacci numbers and the Golden Number

If we take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, ..) and we divide each by the number before it, we will find the following series of numbers:

¹/₁ = 1, ²/₁ = 2, ³/₂ = 1·5, ⁵/₃ = 1·666..., ⁸/₅ = 1·6, ¹³/₈ = 1·625, ²¹/₁₃ = 1·61538...

It is easier to see what is happening if we plot the ratios on a graph:

The ratio seems to be settling down to a particular value, which we call the golden ratio or the golden number. It has a value of approximately 1·618034 , although we shall find an even more accurate value on a later page [this link opens a new window] .

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

The golden ratio 1·618034 is also called the golden section or the golden mean or just the golden number. It is often represented by a greek letter Phi . The closely related value which we write as phi with a small "p" is just the decimal part of Phi, namely 0·618034.

Fibonacci Rectangles and Shell Spirals

We can make another picture showing the Fibonacci numbers 1,1,2,3,5,8,13,21,.. if we start with two small squares of size 1 next to each other. On top of both of these draw a square of size 2 (=1+1).

We can now draw a new square - touching both a unit square and the latest square of side 2 - so having sides 3 units long; and then another touching both the 2-square and the 3-square (which has sides of 5 units). We can continue adding squares around the picture, each new square having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, we will call the Fibonacci Rectangles.

Here is a spiral drawn in the squares, a quarter of a circle in each square. The spiral is not a true mathematical spiral (since it is made up of fragments which are parts of circles and does not go on getting smaller and smaller) but it is a good approximation to a kind of spiral that does appear often in nature. Such spirals are seen in the shape of shells of snails and sea shells and, as we see later, in the arrangment of seeds on flowering plants too. The spiral-in-the-squares makes a line from the centre of the spiral increase by a factor of the golden number in each square. So points on the spiral are 1.618 times as far from the centre after a quarter-turn. In a whole turn the points on a radius out from the center are 1.618⁴ = 6.854 times further out than when the curve last crossed the same radial line.

Cundy and Rollett (Mathematical Models, second edition 1961, page 70) say that this spiral occurs in snail-shells and flower-heads referring to D'Arcy Thompson's On Growth and Form probably meaning chapter 6 "The Equiangular Spiral". Here Thompson is talking about a class of spiral with a constant expansion factor along a central line and not just shells with a Phi expansion factor.

Below are images of cross-sections of a Nautilus sea shell. They show the spiral curve of the shell and the internal chambers that the animal using it adds on as it grows. The chambers provide boyancy in the water. Click on the picture to enlarge it in a new window. Draw a line from the center out in any direction and find two places where the shell crosses it so that the shell spiral has gone round just once between them. The outer crossing point will be about 1.6 times as far from the centre as the next inner point on the line where the shell crosses it. This shows that the shell has grown by a factor of the golden ratio in one turn.
On the poster shown here, this factor varies from 1.6 to 1.9 and may be due to the shell not being cut exactly along a central plane to produce the cross-section.

Here are some more posters available from AllPosters.com that are great for your study wall or classroom or to go with a science project. Click on the pictures to enlarge them in a new window.

Nautilus Wampler, Sondra Buy this Art Print at AllPosters.com

Nautilus Shell Myers, Bert Buy this Art Print at AllPosters.com

Nautilus Schenck, Deborah Buy this Art Print at AllPosters.com

The curve of this shell is called Equiangular or Logarithmic spirals and are common in nature, though the 'growth factor' may not always be the golden ratio.

Reference

The Curves of Life Theodore A Cook, Dover books, 1979, ISBN 0 486 23701 X.
A Dover reprint of a classic 1914 book.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

Fibonacci Numbers, the Golden Section and Plants

One plant in particular shows the Fibonacci numbers in the number of "growing points" that it has. Suppose that when a plant puts out a new shoot, that shoot has to grow two months before it is strong enough to support branching. If it branches every month after that at the growing point, we get the picture shown here.

A plant that grows very much like this is the "sneezewort": Achillea ptarmica.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

Petals on flowers

On many plants, the number of petals is a Fibonacci number:
buttercups have 5 petals; lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals.
The links here are to various flower and plant catalogues:

• the Dutch Flowerweb's searchable index called Flowerbase.
• The US Department of Agriculture's Plants Database containing over 1000 images, plant information and searchable database.

Fuchsia

Pinks

Lily

Daisies available as a poster at AllPosters.com

3 petals: lily, iris
Mark Taylor (Australia), a grower of Hemerocallis and Liliums (lilies) points out that although these appear to have 6 petals as shown above, 3 are in fact sepals and 3 are petals. Sepals form the outer protection of the flower when in bud. Mark's Barossa Daylilies web site (opens in a new window) contains many flower pictures where the difference between sepals and petals is clearly visible.
4 petals Very few plants show 4 petals (or sepals) but some, such as the fuchsia above, do. 4 is not a Fibonacci number! We return to this point near the bottom of this page.
5 petals: buttercup, wild rose, larkspur, columbine (aquilegia), pinks (shown above)
The humble buttercup has been bred into a multi-petalled form.
8 petals: delphiniums
13 petals: ragwort, corn marigold, cineraria, some daisies
21 petals: aster, black-eyed susan, chicory
34 petals: plantain, pyrethrum
55, 89 petals: michaelmas daisies, the asteraceae family.
Some species are very precise about the number