UM - Shroud of Turin

The Shroud of Turin is reputedly Christ's burial cloth. It has been a religious relic since the Middle Ages. To believers it was divine proof the Christ was resurrected from the grave, to doubters it was evidence of human gullibility and one of the greatest hoaxes in the history of art.
No one has been able to prove that it is the burial cloth of Jesus of Nazareth, but its haunting image of a man's wounded body is proof enough for true believers


The Shroud of Turin, as seen by the naked eye, is a negative image of a man with his hands folded. The linen is 14 feet, 3 inches long and 3 feet, 7 inches wide. The shroud bears the image of a man with wounds similar to those suffered by Jesus.
The shroud is wrapped in red silk and kept in a silver chest in the Chapel of the Holy Shroud in the Cathedral of St. John the Baptist in Turin, Italy since 1578.
The shroud is unquestionably old. Its history is known from the year 1357, when it surfaced in the tiny village of Lirey, France. Until recent reports from San Antonio, most of the scientific world accepted the findings of carbon dating carried out in 1988. The results said the shroud dated back to 1260-1390, and thus is much too new to be Jesus' burial linen.
The section of the shroud showing the face reveals dramatic features
when viewed as negative image.Here is an artistic impression of what the face
matching the image might have looked like. This frontal image (above) shows the forearms, wrist, and hands. There appears to be a large puncture wound on the wrist. This is significant because if nails were placed through the palms of the hand, this would not provide sufficient support to hold the body to the cross and tearing of the hands would occur. Only if the nails were placed through the wrists would this provide sufficient support to hold the body fixed to the cross.
We can also see a large blood stain and elliptical wound on the person's right side (remember, in a negative imprint left and right are reversed). From studying the size and shape of this wound and historical records, we can deduce that this wound could have been caused by a Roman Lancea. This lance is pictured in Slide 13.
In addition, by measuring the angle of dried blood on the wrist, one can reconstruct the angle at which this person hung from the cross. He mainly hung from a position 65 degrees from the horizontal. But there is another angle of dried blood at 55 degrees. This shows that this person tried to lift himself up by 10 degrees. Why? Medical studies show that if a person just hangs from a position of 65 degrees in would start to suffocate very quickly. Only if he could lift himself up by about 10 degrees would he be able to breathe. Thus he would have to raise himself up by this 10 degrees by pushing down on his feet which would have to have been fixed to the cross. He would then become exhausted and fall down again to the 65 degree position. Thus, he would continue to shift from these two agonizing positions throughout crucifixion. That is why the executioners of crucifixion would break the legs of their victims to speed up death. If they could not lift themselves up to breathe, they would suffocate very quickly.



The following image shows the most likely position in which Jesus died. This body
position is based on interpretation of the blood stains contained in the shroud.

Unsolved Mysteries - Fibonacci numbers in nature

The sequence, in which each number is the sum of the two preceding numbers is known as the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, ... (each number is the sum of the previous two).
The ratio of successive pairs tends to the so-called golden section
(GS) - 1.618033989 . . . . . whose reciprocal is 0.618033989 . . . . . so that we have 1/GS = 1 + GS.
The Fibonacci sequence, generated by the rule f1 = f2 = 1 , fn+1 = fn + fn-1,
is well known in many different areas of mathematics and science.
However, it is quite amazing that the Fibonacci number patterns occur so frequently in nature ( flowers, shells, plants, leaves, to name a few) that this phenomenon appears to be one of the principal "laws of nature".

Plants do not know about this sequence - they just grow in the most efficient ways. Many plants show the Fibonacci numbers in the arrangement of the leaves around the stem. Some pine cones and fir cones also show the numbers, as do daisies and sunflowers. Sunflowers can contain the number 89, or even 144. Many other plants, such as succulents, also show the numbers. Some coniferous trees show these numbers in the bumps on their trunks. And palm trees show the numbers in the rings on their trunks.

Why do these arrangements occur? In the case of leaf arrangement, or phyllotaxis, some of the cases may be related to maximizing the space for each leaf, or the average amount of light falling on each one. Even a tiny advantage would come to dominate, over many generations. In the case of close-packed leaves in cabbages and succulents the correct arrangement may be crucial for availability of space.

So nature isn't trying to use the Fibonacci numbers: they are appearing as a by-product of a deeper physical process. That is why the spirals are imperfect.
The plant is responding to physical constraints, not to a mathematical rule.
The basic idea is that the position of each new growth is about 222.5 degrees away from the previous one, because it provides, on average, the maximum space for all the shoots. This angle is called the golden angle, and it divides the complete 360 degree circle in the golden section, 0.618033989 . . . .

If we call the golden section GS, then we have
1 / GS = GS / (1 - GS) = 1.618033989 . . . .
If we call the golden angle GA, then we have
360 / GA = GA / (360 - GA) = 1 / GS.

Below there are some examples of the Fibonacci seqeunce in nature. Petals on flowers*

Probably most of us have never taken the time to examine very carefully the number or arrangement of petals on a flower. If we were to do so, we would find that the number of petals on a flower, that still has all of its petals intact and has not lost any, for many flowers is a Fibonacci number:

• 3 petals: lily, iris
• 5 petals: buttercup, wild rose, larkspur, columbine (aquilegia)
• 8 petals: delphiniums
• 13 petals: ragwort, corn marigold, cineraria,
• 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 of petals they have - e.g. buttercups, but others have petals that are very near those above, with the average being a Fibonacci number.


Human Hand

Every human has two hands, each one of these has five fingers, each finger has three parts which are separated by two knuckles. All of these numbers fit into the sequence. However keep in mind, this could simply be coincidence. Human Face

Knowledge of the golden section, ratio and rectangle goes back to the Greeks, who based their most famous work of art on them: the Parthenon is full of golden rectangles. The Greek followers of the mathematician and mystic Pythagoras even thought of the golden ratio as divine.

Later, Leonardo da Vinci painted Mona Lisa's face to fit perfectly into a golden rectangle, and structured the rest of the painting around similar rectangles.

Mozart divided a striking number of his sonatas into two parts whose lengths reflect the golden ratio, though there is much debate about whether he was conscious of this. In more modern times, Hungarian composer Bela Bartok and French architect Le Corbusier purposefully incorporated the golden ratio into their work.

Even today, the golden ratio is in human-made objects all around us. Look at almost any Christian cross; the ratio of the vertical part to the horizontal is the golden ratio. To find a golden rectangle, you need to look no further than the credit cards in your wallet.
Despite these numerous appearances in works of art throughout the ages, there is an ongoing debate among psychologists about whether people really do perceive the golden shapes, particularly the golden rectangle, as more beautiful than other shapes. In a 1995 article in the journal Perception, professor Christopher Green,
of York University in Toronto, discusses several experiments over the years that have shown no measurable preference for the golden rectangle, but notes that several others have provided evidence suggesting such a preference exists.
Regardless of the science, the golden ratio retains a mystique, partly because excellent approximations of it turn up in many unexpected places in nature. The spiral inside a nautilus shell is remarkably close to the golden section, and the ratio of the lengths of the thorax and abdomen in most bees is nearly the golden ratio. Even a cross section of the most common form of human DNA fits nicely into a golden decagon. The golden ratio and its relatives also appear in many unexpected contexts in mathematics, and they continue to spark interest in the mathematical community.
Dr. Stephen Marquardt, a former plastic surgeon, has used the golden section, that enigmatic number that has long stood for beauty, and some of its relatives to make a mask that he claims is the most beautiful shape a human face can have. The Mask of a perfect human faceEgyptian Queen Nefertiti (1400 B.C.)



An artist's impression of the face of Jesus based on the Shroud of Turin and corrected
to match Dr. Stephen Marquardt's mask