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From the article:

The controversy over the carbon-dating evidence with respect to the Shroud centers on the validity of tests performed on three samples snipped from it in April 1988. When the results of testing by the three international laboratories selected to run the newly refined accelerated mass spectrometry method of carbon-dating were made public, all three labs concurred: The Shroud dates sometime between 1260 and 1390 A.D.

I find it interesting that this article does not mention that the 1988 C-14 tests were completely FALSIFIED in three different peer-reviewed scientific articles which demonstrated that although the C-14 tests were accurate on what was tested, the tests failed at the very beginning by the breaking of the test protocols when the sample was taken!

Contrary to what was quoted in the article above, there were NOT three samples taken from the Shroud of Turin, but only ONE, and that one was taken from an area of the Shroud that had been identified by the 1978 Shroud of Turin Research Project scientists as the one area of the Shroud that was NOT homogenous with the main body of the Shroud in that it fluoresced under ultra-violet light while the main body did not! In other words, that part of the Shroud was physically and chemically different from the main body of the Shroud.

The original C-14 agreed on protocols were that EIGHT samples were to be cut from the Shroud from EIGHT separate areas, including both image and non-image sections of the Shroud. At the very last minute those agreed protocols were tossed out by a single person.

That person cut a SINGLE piece out of the Shroud and that piece was then cut into SIX pieces, one of which was discarded because of "colored threads were observed" and it was considered contaminated. The remaining became the C-14 test sub-samples.

Testing done by three different scientists in 1994-2005 proved that the part of the Shroud from where the 1988 C-14 test sample was cut was actually a skillfully rewoven PATCH which included threads from the original Shroud LINEN, made from FLAX, and PATHED in threads made of COTTON. The main body of the Shroud is pure LINEN made from FLAX. The primary rule of any C-14 test is to be sure that what you are testing is homogenous with what you are intending to test. The test failed right there!

These proofs include

Statistical analysis of the 1988 C-14 results from the three lab's results on the FOUR sub-samples showing that the results could not possibly have come from a homogenous sample. . . indicating that even in the original single main sample that had been cut into FIVE sub-samples, were not homogenous, i.e., made of similar aged materials as not ONE of the results—which actually spanned 180 years between the earliest and latest possible creation dates—overlapped any other test's results range of confidence. This statistical work was done by a well respected statistical mathematician who published in a peer-reviewed mathematical journal. That should have been a huge red flag for the testing referee. Instead, the referee merely impermissibly averaged the resultsy a well respected statistical mathematician who published in another peer-reviewed journal.
Chemical tests of threads taken from the surviving control sub-sample of the 1988 C-14 test by one scientist.
Microphotoscopic physical optical examination of the threads from the same source by an independent scientist published in another Scientific Journal showing the threads were from different species of plant demonstrating older, original material made of linen from flax, and newer, dyed Cotton.

THREE SEPARATE approaches conclusively FALSIFIED the 1988 C-14 Tests.

Here is a repost of a commentary I wrote several years ago that explains it in detail.

In 2004, Dr. Raymond Rogers conclusively proved... and had his work successfully peer-reviewed, found accurate, and published in prestigious scientific journals (which is something McCrone has NOT done)... that the sample used in the 1988 carbon dating was inconsistent with the main body of the shroud. Other scientists working from a different direction came to the same conclusion. The samples were NOT physically or chemically the same as the main body of the Shroud... ergo the 1988 C14 testing is proved invalid.

Dr. Rogers worked with photomicrographs and the remaining control sample retained from the five sub-samples cut from the original cutting from the Shroud.

I submit you haven't the foggiest idea what you are talking about. You have heard and repeated something that is untrue.

Chemist and pyrologist Raymond N. Rogers, (Sandia National Laboratory, University of California) and, independently, Dr. John L. Brown, (former Principal Research Scientist at the Georgia Tech Research Institute's Energy and Materials Sciences Laboratory, Georgia Institute of Technology), have done other research and tests and presented evidence in peer reviewed scientific journals that proved that:

The 1988 Carbon 14 Tests were accurate at the current state of the art - on what they tested.
The established, agreed sampling protocols were violated. This is well documented and is beyond contention. The sample cut from the Shroud came from only one area in contravention of the previously agreed protocols which required 8 samples from 8 areas. Instead a single sample from a single area was taken.
The sample that was taken was also taken from the one area all involved scientists had agreed should be avoided as it showed the most dirt and handling damage.
Another reason the area had been excluded as a sampling area was that it was the one area of the Shroud that generally fluoresced under ultra-violet light, indicating a non-similarity to the main body of the shroud which did not fluoresce.
The sample was cut from the corner of the Shroud where the "Raes sample" had been cut 14 years before.
The sample was approximately 1 cm by 7 cm in length and was cut parallel to the long side of the Shroud.
Approximately “…1 cm of the new sample had to be discarded because of the presence of different colored threads that were not similar to the main body of the shroud.” (Where did these foreign threads, interwoven into the sample, come from? - Swordmaker)
Five sub-samples of approximately 1cm x 1cm were cut from the remains of the single original sample cut from the shroud. (For clarity and understanding, let's designate them A to E alphabetically, from the selvage toward the center of the shroud).
The primary sample and the sub-samples were micro-photographed before being packaged and sent for testing.
Sub-samples A and E were sent to the Arizona C14 Lab, B went to Oxford, D to Zurich, and C was retained as a control for future investigation and was untested.
The sub-samples, although chemically cleaned were not microscopically examined or chemically tested, nor were the fibers compared to fibers from other areas of the Shroud by any of the labs.
The C14 Tests were completed and returned results that suggested an origin date for the flax that was in the cloth of 1260 to 1390 AD, with a degree of accuracy of plus or minus ~25 years on each sample.
This spread of possible origin dates of 180 years (1260 minus 25 to 1390 plus 25) should have raised a red flag as the material was supposed to be homogenous and should have all tested within a plus or minus ~75 year spread. In fact, none of the samples' range of confidence overlapped the range of confidence of another in a manner that statistically would indicate the samples were homogenous. This strongly suggested that the samples were, in fact, not homogenous.
Sample A tested younger than sample B which tested younger than Sample D which tested younger than Sample E. The closer the sample was to the center of the Shroud, away from the selvage, the older it tested.
Sample A and Sample E, the samples with both the youngest and oldest reported ages were both tested by the Arizona Lab.
Post C-14 testing and examination of microphotographs of the Primary sample showed a faint demarcation area running somewhat diagonally from the right side of the selvage end (A) to the leftward side of the sample closest to the main body of the shroud (E).
Examination of threads from the retained sample (C) show that threads on the left side of the sample have an "S" twist.
Examination of threads from sample (C) show that threads on the Right side of the sample have a "Z" twist.
Examination of threads taken from main body of the Shroud all have a "Z" twist.
Examination of threads from sample (C) show that threads on the left side of the sample are somewhat (3-5%) thinner in diameter, on average, than threads from the average thread thickness of sample's right half or from the body of the Shroud.
Examination of threads from sample (C) show that threads on the left side of the sample have Cotton intertwined with the Flax.
Examination of threads from right half of the sample (C) and from the main body of the Shroud have no Cotton intertwined with the Flax.
Examination of threads from the retained sample (C) show that threads on the Left side of the sample are encrusted with a plant gum containing alizarin dye extracted from Madder Root, a technique developed in 16th Century France.
Examination of threads from sample (C) show that threads on the Right side of the sample and from the main body of the Shroud are not encrusted with the dyestuff.
Examination of threads from sample (C) show that threads on the Left side of the sample contains up to 2% Aluminum. Chemical testing shows this Aluminum is from Alum (hydrous aluminum oxide), used after the 16th Century as a mordant, a drying agent for retting of cloth.
Examination of threads from sample (C) show that threads on the Right side of the sample and threads from the main body of the Shroud contain no Aluminum.
Chemical testing of threads from sample (C) show that threads on the Left side of the sample Flax's Lignin shows significant levels of Vanillin (> 40%).
Chemical testing of threads from sample (C) show that threads on the Right side of the sample and threads from the main body of the Shroud contain no Vanillin... indicating an age greater than 1300 years.

From an article in Thermochimica Acta: "A linen produced in A.D. 1260 would have retained about 37% of its vanillin in 1978. The Raes threads, the Holland cloth [the shroud's backing cloth], and all other medieval linens gave {positive results from] the test for vanillin wherever lignin could be observed on growth nodes. The disappearance of all traces of vanillin from the lignin in the shroud indicates a much older age than the radiocarbon laboratories reported."
Microscopic examination of the slightly diagonal demarcation area of the sample (C) shows spliced threads, clearly delineating the changes from Left to Right sides of the sample.
Skillful weavers in Europe in the 16th Century used a technique now called French Invisible Reweaving to repair tapestries and arras cloths. Contemporary reports state the method was close to "magical" in the ability to repair damaged cloth. This technique involved spinning and dying thread to closely match the original, splicing the new threads into old threads on the cloth, and reweaving the newly extended threads into the material to match the weaving of the original.
The diagonal demarcation line on the original sample is located so that sample (A)'s suspect (non-similar) threads compose approximately 60% of the sample material. Sample (B)'s suspect (non-similar) threads compose approximately 55% of the sample material. Sample (C)'s, 50% (non-similar) observed and tested. Sample (D)'s, 45%. And Sample (E)'s, (non-similar) 40%. Conversely, threads similar to the main body compose the following approximate percentages of the samples from A to E: 40%, 45%, 50%, 55%, and 60%.
The Shroud underwent repairs after the severe damage from the fire in 1532. Perhaps the corner where the Raes and 1988 C14 test samples were taken was also repaired.
Harry Gove, the inventor of the nuclear accelerator technique that was used to carbon date the Shroud, when asked "How old would a the polluting material have to be to skew the C-14 date of material known to be 1530 AD to show an tested age of 1350 if the polluting material composed 50% of the sample by weight?" He did some calculations and stated, "First Century, give or take 100 years."

The conclusion of the peer reviewed article in Thermochimica Acta states:

"The combined evidence from chemical kinetics, analytical chemistry, cotton content, and pyrolysis/ms proves that the material from the radiocarbon area of the shroud is significantly different from that of the main cloth. The radiocarbon sample was thus not part of the original cloth and is invalid for determining the age of the shroud."

Thus, the 1988 Carbon 14 Testing has been invalidated because the person who took the sample, literally, at the last hour, changed the agreed sampling protocols and took the sample from an area that had been patched, probably in 1532, with contemporary prepared linen thread that had been spun on a spinning wheel that had also spun cotton, then retted with alum, and dyed with alizarin dye from madder root, all done with 15th century technology. The tests were accurate for what they tested: a melange of old and newer material that gave the reported a date that is inaccurate for both the old and the new. It is merely coincidence that the false date of the combined old and new happened to coincide with the first display of the Shroud in Lirey France. The repaired area is not the same as the main body of the shroud and tests are invalid.

New C14 testing should be allowed because there are now a lot of loose samples available since the ill advised "restoration" where they cut away the burned edges around the scorches from the 1532 fire.

I can tell you that there was an unauthorized C-14 test done on one of the threads taken during the 1978 STURP examination and the results were 1st Century, with a degree of confidence of 50 years because the sample was so small.

May I suggest you forget the website you keep repeatedly linking and do some reading of the peer reviewed scientific and scholarly articles on the Shroud? They are mostly available from Barrie Schwortz's website Shroud.com. Barrie was the official photographer of STURP... and he is Jewish. Daniel Porter has put together some of the best and current data in a good popularization of the Shroud information which is more accessible than the scientific papers. Daniel Porter is Freeper Shroudie, and his series of Websites, including Shroudforum.com, Shroud Facts Check, and Shroud Story are an excellent resource based on the latest science.. not on the 30 year old, non-peer-reviewed data from McCrone who ignored agreed protocols on handling Shroud samples which resulted in compromised samples, published his "findings" in a non-peer-reviewed journal (his own vanity publication, "The Microscopist," of which he was both publisher and editor), refused to submit his work for peer-review, and has actually attempted to sabotage other researchers by first, dragging his feet, and then sending them samples that were NOT exemplar of the bulk of samples he had in his possession.