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October 18, 2021

One of the most important Ancient Greek texts is The Sand Reckoner (Ψαμμίτης, Psammites) by Greek mathematician, Archimedes of Syracuse (Ἀρχιμήδης, c.287-c.212 BC). In this work, Archimedes estimates an upper bound for the number of grains of sand that would fill the universe. This calculation involved an early estimate the size of the universe, as well as a notation for extremely large numbers.


Archimedes of Syracuse (c.287-c.212 BC).

Another of the works of Archimedes is his c.250 BC Measurement of a Circle. A portion of this work gives an approximation of the mathematical constant, pi, as follows: 3 and 10/71 < π < 3 and 1/7. This places pi between 3.14084 and 3.14285.

He found these limits by calculating the perimeters of circumscribed and inscribed 96-sided regular polygons.

(Wikimedia Commons image.)

The largest Greek number at that time was the myriad, the word for 10,000 that we use in English to denote a countless number or multitude of a specified item. Archimedes created larger numbers by construction of such objects as myriad myriads (108). Archimedes used the heliocentric model of Aristarchus of Samos, he assumed that the universe was spherical, and that the ratio of the diameter of the universe to the diameter of the Earth's orbit around the Sun was equal to the ratio of the diameter of Earth's orbit around the Sun to the diameter of the Earth.

Plugging in some values for these distances gave a diameter of the universe of just two light years. Archimedes calculated that 1063 grains of sand of about 18.3 μm (0.0183 mm) in diameter would be sufficient to fill it. His estimate for the size of a grain of sand was based on his observation that a poppy seed could contain a myriad (10,000) grains of sand. Modern metrology shows that a small grain of sand is about 62.5 μm (0.0625 mm) in diameter. Large grains of sand are about two millimeters in diameter. The volume of the observable universe is 3.566 x 1080, so I calculate that the universe can contain about 1093 grains of sand.

Sand Reckoner Greek text

This passage at the end of The Sand Reckoner is translated, "I conceive, King Gelon, that among men who do not have experience of mathematics, such a thing might appear incredible...."

It's difficult for anyone to have a working knowledge of everything, and that's a reason why politicians have such limited knowledge of science and mathematics. While most defer to experts for opinions in such subject areas, some are hostile to science and scientists for bringing too much bad news.

The Sand Reckoner was addressed to Gelon II, the King of Syracuse, and Archimedes acknowledged this problem in his time. (Greek text from Ref. 2.[2] English translation from Ref. 3.[3])

Sand appears to be plentiful in the world. Along with gravel and crushed stone, it's one of the most mined materials, used for making concrete and the silicon needed for making integrated circuits. The erosion forces of water and wind create sand grains over geological time scales. While water-eroded sand is suitable for making concrete, the wind-eroded sand of deserts is too smooth for this application. Sand found inland is generally composed of silica (silicon dioxide, SiO2) created from eroded quartz. Beach sand is generally calcium carbonate (CaCO3) created from aragonite. Sand is a non-renewable resource, and about 50 billion tons of sand and gravel are used annually in construction.[5]

A recent review article in the journal, One Earth, examines the idea of sustainable sand use as measured against the sustainable development goals established by the United Nations.[6] Study authors were from McGill University (Montreal, Canada), the University of Copenhagen (Copenhagen, Denmark), the University of Illinois at Urbana-Champaign (Urbana, Illinois), The University of Queensland (Brisbane, Australia), Newcastle University (Newcastle upon Tyne, United Kingdom), the Federal University of Goiás (Goiâum;nia, Brazil), and the London School of Hygiene & Tropical Medicine (London, United Kingdom).[6] They conclude that the practices of the sand industry conflict with eight of the 17 UN goals.[6-7]

Sand dunes in Tadrart Acacus, a desert area in western Libya, part of the Sahara Desert.  2007 photo by Luca Galuzzi

Sand dunes in Tadrart Acacus, a desert area in western Libya, part of the Sahara Desert.

concrete can't be made from desert sand, since the grains are too smooth.

The phrase, "Water, water, everywhere, nor any drop to drink...," from the The Rime of the Ancient Mariner by Samuel Taylor Coleridge (1772-1834) can be re-imagined as "Sand, sand, everywhere, nor any grain to use."

(A 2007 photo by Luca Galuzzi, via Wikimedia Commons.)

While sand is the most used mineral in the world and a main component of modern civilization, there's a limited supply, and nature makes sand at too low a rate for it to be sustainable at the present level of its use.[7] The study focused on low- and middle-income countries, and the research team examined sand production with respect to the United Nations Sustainable Development Goals, a holistic scale of broad topic areas that's independent of the specific resource analyzed.[6] They found that the sand industry conflicts with eight of the 17 of the goals.[6] While I find that several of these goals are too general and not easily quantifiable (e.g., Peace, justice and strong institutions), there are some that demand attention. These are
• Clean water and sanitation
• Responsible consumption and production
• Climate action
• l Life below water
• Life on land

Says study author, Lars Lønsmann Iversen of the University of Copenhagen,
"Sand is the most used resource in the world, next to water, but the way it is extracted is in direct conflict with eight of the 17 UN Sustainable Development Goals. We do not have endless amounts of sand, and within the next 50 years we expect it to be in short supply. It is therefore vital that we are now able to point to significant sustainability issues involving the world's sand resources."[7]

Sand is excavated mostly from mines and along rivers in low- and middle-income countries, and therein lies most of the problem.[7] Such mining at rivers comes with great environmental and local human quality of life impact.[7] Removal of sand along rivers changes their natural boundaries and their natural flow. Sudden removal of sand affects the sediment discharge in these rivers and causes increased erosion and an increased risk of flooding.[7] Drinking water is also affected.[7] Says Iversen,
"The negative consequences are to a large extent due to the fact that sand is an unregulated resource. Unregulated resources can be misused and exploited by rich countries. The human consequences of these activities can be solved by regulating the way the resource is used locally. For example, the miners may suffer from poor working conditions. The precious metal and diamond industries are already focusing on this issue, but in the sand industry it has gone more or less undetected."[7]

These problems must be balanced against the positive aspects of the sand industry. It provides millions of jobs in developing countries, and sand is an important component in the economic development in these developing countries.[7]

Alternatives to sand are being investigated, such as use of plastic instead of sand as an asphalt additive. I wrote about asphalt and other paving materials in an earlier article (Paving Materials, July 13, 2020). Engineered wood can replace concrete in some buildings. Research is also underway on manipulating desert sand to make it a suitable concrete additive.[7]

The Sandman, Ole Lukøje

The Sandman is a mythical character who induces sleep and encourages dreams by sprinkling magical sand onto a person's eyes.

In Scandinavian folklore, the Sandman (as depicted) is called "Ole Lukøje".

The origin of the myth is the sand-like rheum, the crust in the corners of the eyes seen upon waking.

(A depiction of Ole Lukøje by Vilhelm Pedersen, via Wikimedia Commons)


  1. Archimedes, "The Sand Reckoner," Greek text at The Computer Science Laboratory of Ecole Polytechnique (Postscript file).
  2. Archimedes, "The Sand Reckoner," Greek text at the Personal Web Page of Heinrich F. Fleck (PDF file).
  3. Archimedes, "The Sand Reckoner," English translation at the website of Ilan Vardi (Postscript file).
  4. Ilan Vardi, "The Legacy of Archimedes (287-212 B.C.), Departement d'Informatique, Ecole Polytechnique, Palaiseau, France .
  5. Vince Beiser, "Why the world is running out of sand," BBC, November 18, 2019
  6. Mette Bendixen, Lars L. Iversen, Jim Best, Daniel M. Franks, Christopher R. Hackney, Edgardo M. Latrubesse, and Lucy S. Tusting, "Sand, gravel, and UN Sustainable Development Goals: Conflicts, synergies, and pathways forward," One Earth, vol. 4, no. 6 (August 20, 2021), pp. 1095-1111, https://doi.org/10.1016/j.oneear.2021.07.008.
  7. Sand is one of our most used resources, but the industry is not sustainable, University of Copenhagen Press Release, August 31, 2021.

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