Brittlestars form a large group of sea animals that are similar to starfish. There are more than 2,000 species of brittlestars. However, this article will focus on two of them, Ophiocoma pumila (Figure 1a) and Ophiocoma wendtii (Figure 1b). In spite of their similar appearance, these two kinds of brittlestars have one main difference. While O. pumila is insensitive to light, O. wendtii is highly light sensitive. For example, the latter has different colors at day and night, as shown in Figure 1b, left and right respectively. More interestingly, O. wendtii can sense shadows of predators and quickly move into dark areas such as a cave or underneath a rock.

To understand the mechanisms behind the difference in light sensitivities of these two species, Joanna Aizenberg and her colleagues investigated1 the microstructure of both brittlestars’ outer skeletons with an electron microscope and came up with a striking result: The top surface of O. wendtii’s skeleton has very well ordered lens-like hemi spherical elements (Figure 1f). The cross section image of one of those hemispheres actually looks like a compound lens made up from two hemispheres with different diameters (Figure 1g). On the other hand, O. pumila’s skeleton had a typical stereom (sponge-like calcite) structure (Figure 1e). These images strongly suggest that the lenses in O. wendtii’s skeleton are responsible for the relatively high light sensitivity. However, understanding how that really happens require further investigation.

It is well-known that spherical lenses suffer from a problem called “spherical aberration,” which means that the light rays that are closer to the optical axis are focused at a different point than the ones that are away from the axis. A quick solution to this problem is to use two lenses, whose diameters have a certain ratio, back to back; this helps to correct the aberration originating from the first one with the second one. Interestingly, when Aizenberg et. al. calculated1 the optimum compound lens configuration for O. wendtii’s skeleton, which has the minimum aberration, their result matched the original lens structure perfectly (the orange outline in Figure 1e). They were also able to locate the focal point of these lenses (d = 4-7 um* below the lens) with the same method. Their further electron microscopy studies showed optically sensitive nerve bundles exactly at that location. All these results clearly show that O. wendtii’s skeleton has the perfect geometry to collect and focus light to improve its light sensitivity. However, there is one big question about these lenses: their material.

Calcite, a kind of calcium carbonate (CaCO3), is a common ingredient of the shell or the skeleton of marine organisms. Interestingly, the birefringence property of calcite makes it very unfavorable as a lens material. In a birefringent material the speed of the light depends on the direction it travels with respect to the crystallographic axes of the material. As a result, if one looks through it, they will observe a doubly refracted image (Figure 2). Being the most famous example of birefringent crystals, calcite’s refractive index is 1.64 parallel to one crystallographic axis and 1.49 in the perpendicular direction. Therefore a regular calcite lens cannot focus light on a single spot, unless it is oriented along a special crystallographic axis (c-axis to be specific), which would be along the diagonal of the prism in Figure 2.

At this point we are not surprised to learn that the optical axis of the O. wendtii’s lenses, and the c-axis of the calcite crystal that they are made of, indeed overlap. We are not surprised because we already had a strong feeling that these lenses should work. However, it is quite surprising that these little creatures can grow single crystals of calcite with a specific crystallographic orientation. As Kenneth Towe states in the context of a similar study, “This precise orientation of crystals is the big mystery of biomineralization. Organisms know how to do it; we do not yet know how they know.”3

Biomineralization, the controlled deposit of inorganic minerals by living organisms, is a very active research field attracting many scientists from various disciplines, including biology, physics, chemistry, and material science. In general, controlling crystal structures at small length scales is a very challenging task. Scientists spend millions of dollars to build state-of-the-art facilities for single crystal materials synthesis. They work in clean rooms, under an ultra high vacuum and at extremely high temperatures. On the other hand, from brittlestars to large whales, almost all living creatures have biominerals, such as bones and shells, manufactured in chemically dirty environments and at decent temperatures. Organisms are apparently equipped more efficiently than our laboratories are.

A. Ali Eren has a Ph.D. in Physics and lives in the USA. He studies physical chemistry of biological processes.

References

*1 um (micron) is one thousandth of a millimeter. Human hair is approximately 100 micron thick.

1. Aizenberg, Joanna, et al. “Calcitic microlenses as part of the photoreceptor system in brittlestars.” Nature 412.6849 (2001): 819-822.
2. ttp://jademellor.com/blog/2013/6/21/rainbow-rhombus, accessed 4/20/2014
3. Towe, Kenneth M. “Sea urchins as crystallographers.” Science 311.5767 (2006): 1554-1555.

Mathematics, or “the queen of the sciences” as Carl Friedrich Gauss called it, plays an important role in our lives. A world without mathematics is unimaginable. Throughout history, many scholars have made important contributions to this science, among them a great number of Muslims. It is beyond the scope of a short article like this one to mention all the contributions of Muslim scholars to mathematics; therefore, I will concentrate on only four aspects: translations of earlier works, and contributions to algebra, geometry, and trigonometry. In order to understand fully how great were the works of scholars in the past, one needs to look at them with the eye of a person of the same era, since things that are well-known facts today might not have been known at all in the past.

There has never been a conflict between science and Islam. Muslims understand everything in the universe as a letter from God Almighty inviting us to study it to have knowledge of Him. In fact, the first verse of the Qur’an to be revealed was:

Read! In the Name of your Lord, Who created… (Alaq 96:1).

Besides commanding us to read the Qur’an, by mentioning the creation the verse also draws our attention to the universe. There are many verses which ask Muslims to think, to know, to learn and so on. Moreover, there are various sayings of the Prophet Muhammad, peace be upon him, encouraging Muslims to seek knowledge. One hadith says, “A believer never stops seeking knowledge until they enter Paradise” (al-Tirmidhi).

In another hadith, the Prophet said, “Seeking knowledge is a duty on every Muslim” (Bukhari). Hence it is no surprise to see early Muslim scholars who were dealing with different sciences.

Translations

Prophet Muhammed (pbuh) said, “Knowledge is the lost property of a Muslim; whoever finds it must take it” [1]; hence Muslims started seeking knowledge. One way they did this was to start translating all kinds of knowledge that they thought to be useful. There were two main sources from which Muslim scholars made translations in order to develop the field of science, the Hindus and the Greeks. The Abbasid caliph al-Mamun (804–832) had a university built and ordered its scholars to translate into Arabic many works of Greek scholarship. Between 771 and 773 CE the Hindu numerals were introduced into the Muslim world as a result of the translation of Sithanta from Sanskrit into Arabic by Abu Abdullah Muhammad Ibrahim al-Fazari. Another great mathematician, Thabit ibn Qurra, not only translated works written by Euclid, Archimedes, Apollonius, Ptolemy and Eutocius, but he also founded a school of translation and supervised many other translations of books from Greek into Arabic. While Hajjaj bin Yusuf translated Euclid’s Elements into Arabic, al-Jayyani wrote an important commentary on it which appears in the Fihrist (Index), a work compiled by the bookseller Ibn an-Nadim in 988. A simplified version of Ptolemy’s Almagest appears in Abul-Wafa’s book of Tahir al-Majisty and Kitab al-Kamil. Abu’l Wafa Al-Buzjani commented on and simplified the works of Euclid, Ptolemy and Diophantus. The sons of Musa bin Shakir also organized translations of Greek works.

These translations played an important role in the development of mathematics in the Muslim world. Moreover, the ancient Greek texts have survived thanks to these translations.

Algebra and geometry

The word “algebra” comes from “Al-Jabr”, which is taken from the title of the book Hisab Al-Jabr wal Muqabala by Muhammad ibn Musa al-Khwarizmi (780–850). Al-Khwarizmi, after whom the “algorithm” is named, was one of the great mathematicians of all times. Europe was first introduced to algebra as a result of the translation of Khwarizmi’s book into Latin by Robert Chester in 1143. The book has three parts. The first part deals with six different types of equations:

(ax² = bx) ; (ax² = b) ; (ax = b) ; (ax² + bx = c) ; (ax² + c = bx) ; (bx + c = ax²)

Khwarizmi gives both arithmetic and geometric methods to solve these six types of problems [2]. He also introduces algebraic multiplication and division. The second part of Hisab Al-Jabr deals with mensuration. Here he describes the rules of computing areas and volumes. Since Prophet Muhammad, peace be upon him, said, “Learn the laws of inheritance and teach them to people, for that is half of knowledge,”[3] the last and the largest part of this section concerns legacies, which requires a good understanding of the Islamic laws of inheritance. Khwarizmi develops Hindu numerals and introduces the concept of zero, or “sifr” in Arabic, to Europe. The word “zero” actually comes from Latin “zephirum,” which is derived from the Arabic word “sifr.”

The three sons of Musa bin Shakir (about 800–860) were perhaps the first Muslim mathematicians to study Greek works. They wrote a great book on geometry, Kitab Marifat Masakhat Al-Ashkal (The Book of the Measurement of Plane and Spherical Figures), which was later translated into Latin by Gerard of Cremona. In the book, although they used similar methods to those of Archimedes, they move a step further than the Greeks to consider volumes and areas as numbers, and hence they developed a new approach to mathematics. For example, they described the constant number pi as “the magnitude which, when multiplied by the diameter of a circle, yields the circumference.”[4]

A well-known poet, philosopher and astronomer Omar Khayyam (1048–1122) was at the same time a great mathematician. His most famous book on algebra is Treatise on the Demonstration of Problems of Algebra. In his book besides giving both arithmetic and geometric solutions to second degree equations he also describes geometric solutions to third degree equations by the method of intersecting conic sections. He also discovered binomial expansion [26]. His work later helped develop both algebra and geometry.

Thabit bin Qurra (836–901) was an important mathematician who made many discoveries in his time. As mentioned in the Dictionary of Scientific Biography [5] he “played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-Euclidean geometry. In astronomy Thabit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics.”

To give an idea of his importance, we will just give here, without details, one of his theorems on amicable numbers. Two natural numbers m and n are called “amicable” if each is equal to the sum of the proper divisors of the other:

for n > 1, let pn=3.2^2n–1 and qn=9.2^2n–1–1. If p_n–1 , p_n and q_n are prime numbers, then a=2n p_n–1 p_n and b=2ⁿq_n are amicable. [6]

Abu Kamil (about 850–930), an Egyptian mathematician, wrote the Book on Algebra which consists of three parts:

(1) Solutions of quadratic equations,

(2) Application of algebra to geometry,

(3) Diophantine equations.[7],[8]

He improved the work of Khwarizmi and applied algebraic methods to geometry. His research was on quadratic equations, multiplication and division of algebraic quantities. His work also includes addition and subtraction of radicals. He found the following formulas:

ax.bx=abx²; a(bx)=(ab)x; (10–x)(10–x)=100+x²–20x

Abu Kamil also wrote the Book On Surveying and Geometry, which was intended for government land surveyors. There, he stated the nontrivial rules for calculating areas, volumes, perimeters, and diagonals of different objects in geometry.[9]

Ibrahim ibn Sinan (908–946), a grandson of Thabit bin Qurra, was both an astronomer and a mathematician. Fuat Sezgin writes, “He was one of the most important mathematicians in the medieval Islamic world.” [10] He studied geometry, and his work on calculation of the area under the graph of a parabola is especially appreciated. Going further than Archimedes, he introduced a more general method of integration. [11]

Abu Bakr ibn Muhammad ibn al-Husayn al-Karaji (953–1029), also known as al-Karkhi, is regarded as the first person to have developed algebraic operations without using geometry. One of his major works was Al-Fakhri fi’l-jabr wa’l-muqabala (Glorious on algebra). Historian Woepcke recognizes Al-Fakhri as the beginning of the theory of algebraic calculus. [12] Here, al-Karkhi introduced the monomials x, x², x³, … and 1/x, 1/x², 1/x³, … and explained product rules among them. Moreover, he was the first to find the solutions of the equations ax²ⁿ+bxⁿ=c. [13] Al-Karkhi proved the sum formula for integral cubes by using the method of proof by induction, and hence became the first to use this method. [14]

Abu’l Hasan ibn Ali al-Qalasadi (1412–1486) was an Andalusian Muslim mathematician. His main contribution was to introduce algebraic symbolism, and he used short Arabic words for his symbols. For example, he used the symbol for the sound “sh” from the Arabic word meaning “thing” to represent what we call x, the unknown. [15]

Trigonometry

Khwarizmi also contributed to trigonometry. He established accurate trigonometric tables for sine and cosine, and he was the first to introduce tangent tables. [16] In 1126, these works were translated into Latin by Adelard of Bath.

Al-Battani or Albetagnius (about 850–929) was a Muslim astronomer and mathematician. In his research on astronomy he used trigonometric methods which were a lot more advanced than the geometric methods used by Ptolemy. [17] He introduced trigonometric ratios. For example, for a right triangle with adjacent sides a and b, he gives the formula b sin(A) = a sin(90⁰ – A), which is equivalent to tan A = a/b. He was the first to introduce the cotangent function. [18]

Muhammad Abu’l Wafa (940–998), born at Buzjan in Khorasan, introduced the use of secant, cosecant and tangent functions. He gave a new method of constructing sine tables. He calculated sin(30^0) with an accuracy of up to eight decimal digits. He improved spherical trigonometry and proved the law of sines for general spherical triangles. [19] In particular, he developed the half/double angle formulas:

2 sin² (x/2)=1–cos x; sin 2x=2sin x cos x

He was the first to introduce the notion of secant and cosecant, and hence completed the list of all six trigonometric functions. [20]

Abu Abd Allah Muhammad ibn Muadh Al-Jayyani (989–1079) was an Arab mathematician from Andalus. He was the author of The Book of Unknown Arcs of a Sphere which was “the first treatise on spherical trigonometry.” [21] Here he mentioned formulas for right handed triangles and law of sines. He also stated the formula for the solution of a spherical triangle in terms of the polar triangle. [21] He had a strong influence on the West.

Another outstanding mathematician Nasir al-Din al-Tusi (1201–1274) wrote Treatise On The Quadrilateral, considered the best book on trigonometry written in medieval times, [25] later translated into French by Alexandre Carathéodory Pasha in 1891. In his book al-Tusi made enormous advances in plane and spherical trigonometry. The Dictionary of Scientific Biography [22] states, “This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth.” The well-known sine law is also stated in this work: a/sin A = b/sin B = c/sin C.

Ghiyath al-Din al-Kashi (1393–1449) produces sine tables of up to eight decimal places. In 1424, he computed 2π to an accuracy of sixteen decimal digits. He wrote a very impressive book on mathematics: Miftah al-Hussab (Key to Arithmetic). His main purpose in this book is to provide sufficient knowledge of mathematics for those who are working on astronomy,surveying, architecture, accounting and trading. [23] He also describes how to find the fifth root of any number. [24]

Unfortunately, the contributions of Muslims often go unrecognized. Muslim scholars contributed to science in many aspects such as mathematics, astronomy, geography, philosophy, medicine, art, architecture and so on. However, today few realize that in that era Islam played an important role in all aspects of life. Europe faced losing the works of major scholars, but as a result of their translations into Arabic most of this scholarship not only survived, but was further developed. Inspired by the Qur’an and hadiths, Muslims sought knowledge for the benefit of humankind. As the Qur’an says, “Are those who know equal to those who know not?”(Zumar 39:9). We should appreciate the scholars of all eras for their contributions to science.

Shirali Kadyrov is a PhD candidate at the Ohio State University, Mathematics Department.

References

1. Tirmidhi, `Ilm, 19.

2. B.L. van der Waerden, A History of Algebra.

3. Ibn Maja, Hadith No: 2719.

4. D. El-Dabbah, The geometrical treatise of the ninth-century Baghdad mathematicians Banu Musa (Russian), in History Methodology Natur. Sci., No. V, Math. Izdat. (Moscow, 1966), 131–139.

5. Y. Dold-Samplonius, A. T. Grigorian, B. A. Rosenfeld, Biography in Dictionary of Scientific Biography (New York 1970–1990).

6. For more, see S. Brentjes and J. P. Hogendijk, Notes on Thabit ibn Qurra and his rule for amicable numbers, Historia Math. 16 (4) (1989), 373–378.

7. R. Lorch, Abu Kamil on the pentagon and decagon, Vestigia mathematica (1993), 215–252.

8. J. Sesiano, La version latine medievale de ‘l’Algebre d’Abu Kamil, in Vestigia mathematica (Amsterdam, 1993), 315–452.

9.J. Sesiano, Le Kitab al-Misaha d’Abu Kamil, Centaurus 38 (1996), 1–21.

10. F. Sezgin, History of Arabic literature (German) Vol. 5 (Leiden, 1974), 292–295.

11. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Ibrahim.html

12. F. Woepcke, Extrait du Fakhri, traite d’Algebre par Abou Bekr Mohammed Ben Alhacan Alkarkhi (Paris, 1853).

13. Boyer, Carl B. (1991). “The Arabic Hegemony”, A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 239. ISBN 0471543977.

14. Victor J. Katz (1998). History of Mathematics: An Introduction, p. 255–259. Addison-Wesley. ISBN 0321016181.

15. J. Samso, Las ciencias de los antiguos en al-Andalus (Madrid, 1992).

16. http://en.wikipedia.org/wiki/History_of_trigonometry

17. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Al-Battani.html

18. http://www.unhas.ac.id/~rhiza/saintis/battani.html

19. http://www.britannica.com/EBchecked/topic/2127/Abul-Wafa

20. http://www.bhatkallys.com/article/article.asp?aid=3442

21. O’Connor, John J. & Robertson, Edmund F., Abu Abd Allah Muhammad ibn Muadh Al-Jayyani.

22. S. H. Nasr, Biography in Dictionary of Scientific Biography (New York 1970–1990).

23. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Al-Kashi.html

24. http://www.bhatkallys.com/article/article.asp?aid=3442

25. http://members.tripod.com/worldupdates/newupdates10/id142.htm

26. Heinrich Dorrie, David Antin (1965). 100 Great Problems of Elementary Mathematics: Their History and Solution, p.34–36. ISBN 0486613488.

Ancient people, considering it very important to determine Earth’shape, derived two important clues from the night skies. According to Aristotle (384-322 bce), these were lunar eclipses and the North Star. Lunar eclipses occur when the sun, Earth, and the moon line up in such a way that Earth temporarily blocks the sun’s light from reaching the moon while its circular shadow gradually crosses the moon’s face. The North Star appears lower in the sky the further south we go: at the Equator it lies directly on the horizon, at a latitude of 45 it is 45 above the horizon; and at the North Pole it is directly overhead. However, it is not visible south of the Equator.(1) As both of these indicate a spherical Earth, the scholars of that time discarded the idea of a flat Earth.

The more challenging question was how to determine Earth’s size. Eratosthenes of Alexandria (third century bce) had a simple yet brilliant idea: insert a gnomon (a vertical stick) into a level piece of ground. This enabled him to determine noon’s exact time (when the shadow was the shortest). It was also used as a compass, for in the Northern Hemisphere the gnomon’s shadow points north.

But can such a simple device determine Earth’s size? Aswan, located about 500 miles south of Alexandria, sits on the Tropic of Cancer. So, at noon of June 21 (the summer solstice), a gnomon inserted there has no shadow. By doing just that in Alexandria, Eratosthenes found that the angle was 1/50 of a circle’s circumference (i.e., 2p/50). In other words, the angle at Earth’s center corresponding to the arc between Aswan and Alexandria on Earth’s surface is 1/50 of a circle’s circumference. Since the distance between Alexandria and Aswan is 500 miles, Earth’s circumference should be 25,000 miles, which is its actual circumference.(2) Thus, Earth’s size and shape was pretty well established over 2,000 years ago.

This knowledge was lost to Europe when the ancient civilizations crumbled. But Islamic civilization and culture, which was rising at roughly the same time as the West was declining, produced scholars and scientists who translated and refined quite a bit of this ancient knowledge. For example, in 1424 al-Kashi used Archimedes’ method of computing to determine its values to 16 decimal places. Ulug Beg compiled the greatest star catalog known at that time. During al-Ma’mun reign (813-833), al-Khwarizmi measured one degree of latitude on Earth’s surface and obtained the result of 57 miles. This means that Earth’s circumference is 360×57 = 20,520 miles.(3) Thus, in the ninth century, Muslim scientists knew that Earth was spherical and had a good idea of its size. Most Europeans at that time, believed that Earth was flat and the universe impenetrable.

The Qur’an describes Earth’s geographical shape and change in that shape: Do they not see how We gradually shrink the land from its outlying borders? Is it then they who will be victors? (21:44).(4) The reference to shrinking could relate to the now-known fact that Earth is compressed at the poles.

At a time when people generally believed that Earth was flat and stationary, the Qur’an explicitly and implicitly revealed that it is round. More unexpectedly still, it also says that its precise shape is more like an ostrich egg than a sphere: After than He shaped Earth like an egg, whence He caused to spring forth the water thereof, and the pasture thereof (79: 30-32).

The verb daha’ means “to shape like an egg,” and its derived noun da’hia is still used to mean “an egg.” As this may have appeared incorrect to pre-modern scientists, some interpreters misunderstood the word’s meaning as “stretched out,” perhaps fearing that its literal meaning would only confuse people. Modern scientific instruments recently established that Earth is shaped more like an egg than a perfect sphere, and that there is a slight flattening around the poles and a slight curving around the Equator.

The West receives “lost” knowledge

An enduring Western myth is that Columbus had to overcome a pervasive belief that he would sail off the edge of a flat Earth by sailing west to Asia. This myth stems in part from compressing the past and conflating the early Middle Ages, when Europe’s belief in a flat Earth was widespread, with the late Middle Ages, when Europe’s knowledge had caught up with and partially surpassed that of ancient Greece and medieval Islam.

During the Renaissance, Europe came into contact with “lost” knowledge by translating Greek and Arabic works. One important book was Ptolemy’s Geography, which accepts Earth’s spherical shape. Geography once more became available in the original Greek, which was not widely known in the thirteenth century. This book was translated into Latin in the late fifteenth century and became widely known. Columbus owned a copy printed in 1479.

By the time of Columbus, the idea of a spherical Earth was widely accepted in theory. Columbus believed this and wanted to sail west to the eastern shores of Asia. Earth’s size was the real issue. Ptolemy’s estimate was as much as 20% too low. Also, he vastly overestimated Asia’s size. The resulting map depicted an Earth with oceans between Europe’s western tip and Asia’s eastern tip, which was well within range of the provisions that ships of that time could carry. Columbus’ estimate of the distance to Asia was wrong, as was his assumption that there was no land between Europe and Asia. Fortunately for him, these two “wrongs” made a “right,” with all of its attendant fame and glory.

The Shape of the Universe

So far, we have given external information (i.e., lunar eclipses and the North Star) about Earth’s spherical shape based upon its position in the universe. If we use this method to determine the universe’s shape, we must observe it in an external manner. As this is not possible, let’s reconsider the question of Earth’s shape with a slight change: Can we determine Earth’s shape by using measurements and observations done only on its surface, and thereby acquire intrinsic information that can inform us of the universe’s shape?

Karl Gauss (1777-1855) answered this question positively by inventing “curvature,” which measures a given surface’s “bumpiness” at a specific point. A flat piece of paper has no bumps and so its curvature is zero. But if we look at a sphere at each point, we see some bumpiness. Gauss called such bumpiness “positive curvature.” Another kind of bumpiness is “saddle-shaped.” We can think of positive curvature at a point as follows: If we put a piece of flat paper on a surface at that point, the surface lies totally on one side of the paper. But in negatively curved space, this cannot happen.

To describe this concept formally (minus some technicalities), assume constant curvatures on the shapes in question. In other words, the shape is totally symmetric and every point has the same amount of bumpiness. There are several ways to describe curvature. Gauss’s formulation for curvature is brilliant. But before that, let’s look at his intrinsic proof for a spherical Earth. Imagine an orchard so large that any deviation from flatness is perceptible. First plant trees on the Equator every 100 kms (the approximate distance between two meridians on the Equator). Then plant another tree 100 kms (the approximate distance between two parallels) north of each tree, and do this several times. If Earth is flat, the distance between them would be same. But since the distance between the two consecutive trees (on the same parallel) decreases, Earth is spherical.

Having seen that an intuitively positive curvature implies a spherical shape, we want to follow this method to get an idea about the universe’s shape. Georg Riemann (1826-66), trying to do just that, invented “curved space” and explained how to compute its curvature. We could launch six probes at equally spaced points along the Equator, and have each of them continually monitor the distance to the two adjacent probes. If space is flat, the distances at any point in its journey would equal the distance from the probe to Earth’s center (an equilateral triangle). For negative curvature, the distance between probes would grow faster than the distance the probe had traveled from Earth; in positively curved space, the distance between probes would grow slower than the distance covered by the probes since leaving Earth.

There are two common misconceptions about the curvature of space. The first one is that curvature is a rather vague or qualitative concept. In reality, it is quite precise and assigns to each point in space and each direction at that point an exact number determined by the shape of the space near the specific location. The second one is that to describe curved space, one must think of it as “curving” into a fourth dimension. This can be useful in visualizing curved space for people familiar with four-dimensional Euclidean space (four-dimensional coordinate space). Unfortunately, science popularizers and science fiction writers often lace this concept with mystical overtones. This is more likely to confuse average people. In other words, measurements made in ordinary three-dimensional space may disagree with the results embodied in Euclidean geometry, for curvature measures the degree and kind of deviation from the Euclidean model.

Riemann also proposed a radically different (non-Euclidian) model for the universe: “spherical space.” This would be the case if space had a constant positive curvature. Based on this, he said that the universe should be a hypersphere (a three-dimensional sphere). The usual sphere is two-dimensional and lives in three-dimensional Euclidean space. In general, n-dimensional sphere is described as in the (n+1)-dimensional Euclidean space, and the set of points whose distance from origin (the point 0) is 1.

The more intuitive way to describe hypersphere comes from the usual sphere. Starting from a point in the sphere called the South Pole, and as we go in a direction in the sphere, we see concentric circles becoming larger until we reach the Equator, after which they become smaller and we finally reach North Pole. The situation is similar in hypersphere. Start from a point in the sphere called the South Pole, and as we go in a direction in the sphere, the concentric “spheres” become larger until we reach the Equator, after which they become smaller until we reach the North Pole. We can generalize this concept for any sphere of any dimension.

Earlier philosophers speculated that the universe was infinite in extent; others (e.g., Plato, Aristotle, Newton, and Leibniz) rejected this as implausible. But the alternative seemed equally dubious: If it did not go on forever, then “like the flat Earth” it had to end somewhere. And, what was beyond that? This model solved the Euclidean paradox of the universe’s “edge,” for if the universe is positively curved, it can be finite in extent and still not have any “edge.” In Riemann’s model, every part of the universe looks just like every other part, as far as shapes and measurements go.

Qur’an 51:47-48 mentions the universe’s spreading out or expansion in space: And the firmament: We constructed it with power and skill, and We are spreading it. This verse reveals that the distance between celestial bodies is increasing, which means that the universe is expanding.

Hubbel’s law

The most surprising discovery of the twentieth century was made by Edwin Hubble in 1929: The universe is not static, but is in a state of rapid expansion. Based upon his observations, he stated Hubble’s Law: Other galaxies are receding from us, the rate at which they recede depends upon their distance, and there is a constant ratio (the Hubble constant) between their velocity and their distance from us.

This law’s most dramatic consequence is what it tells about how we got to where we are now. If distances between galaxies increase as we look toward the future, they must decrease as we go back in time. Each ring of galaxies must have been closer to us in the past; the further away (or back in time) we go, the closer they would have been, and the faster they appear to be moving toward us.

Hubbel’s evidence was limited to a few relatively nearby galaxies. Over the years, however, thousands of observations extended and refined the measurements, and confirmed the general correctness of the velocity-distance relation. Current best estimates are that those galaxies are a billion light-years away (a light-year is roughly 6 trillion miles). Assuming that light always travels at the same speed, those galaxies must have been 1/20 of a light-year closer to us each year in the past. To have ended up a billion light years away, they must have started at exactly the same point as we did “the Big Bang” some 20 billion years ago.

Let’s start by using concentric rings of galaxies at intervals of a billion light-years. Then there are 20 rings, because five rings from us represents galaxies 5 billion light-years distant from us. To see them, we need to see their light that has been traveling for 5 billion years. Thus, we now see their position 5 billion years ago. As there was nothing 20 billion years ago, the outmost ring must the twentieth ring. This might sound paradoxical, as the circles of galaxies seem to grow larger as they move further away from us. However, the paradox is only apparent. Assuming Earth is in the South Pole and that the rings are a sphere’s latitudes, the rings become larger by the Equator and then become smaller until, in the twentieth ring, we reach the North Pole. This time, the rings are spheres and thus fit in the hypersphere. So Hubble’s Law supports our model of hypersphere for the universe.

But how can an expanding universe fit into our picture? In the sphere, the whole surface is expanding, just like inflating a balloon. So the distance from us (at Earth) to the Big Bang is increasing in all directions. In other words, any two points in the universe recede from each other, just as any two points on the balloon recede from each other during inflation.

The issue of time

So far, we have considered the universe’s shape at a fixed time. But, in physics, it is useful to consider space and time together. After Einstein’s brilliant publications about special relativity, Hermann Minkowski (1864-1909) proposed a very useful four-dimensional space-time model as the fabric of the physical universe. In a global picture, each fixed time represents a thin slice of space-time. Like an onion, each layer (assuming there are infinite very thin layers) corresponds to the universe at different fixed times. Given that each fixed time is a hypersphere, the layers are hyperspheres. According to Hubble’s Law, the hypersphere becomes larger as time passes. Just like an onion, the inner layers are smaller and the outer layers are larger.

In an ordinary onion, the layers are usually spheres; in the universe, the layers are hyperspheres. Assuming that the outer-most slice represents the universe at this time, the space-time “so far” is a four-dimensional onion, with layers of the universe at different times. The center of this “onion” corresponds to the Big Bang. We receive the picture of space-time until “this time.” Now, let mathematics predict the future of the space-time, just as Newton’s laws allowed a detailed description of the solar system’s future course. As time evolves, the universe expands, distances between galaxies grow, and gravity weakens. Thus, the space-time curvature diminishes and successive hyperspheres grow at a slower rate.

Two possibilities emerge: The hyperspheres continue to grow indefinitely, although at an ever-decreasing rate, or reach a maximum size and start to contract, in exactly the same fashion as the parallels of latitude on Earth: starting at the North Pole, growing until they reach their maximum size at the Equator, and then begin contracting toward the South Pole. If the universe contracts, distances between galaxies would decrease, gravity and curvature would increase, and the successive hyperspheres would shrink ever faster, eventually contracting into a single point: the “Big Crunch.”

We could then draw a map of the universe as a succession of hyperspheres growing in size for during the first half of its life and contracting during the second half. All space-time would then form a kind of super-hypersphere a four-dimensional object known as a “four-dimensional sphere.”

Conclusion

At a fixed time, the universe should be a hypersphere. When the time dimension is added, space-time should be super-hypersphere, with a “Big Bang” like the South Pole, each fixed time of the universe corresponding to the parallels, and finally a “Big Crunch” corresponding to the North Pole in our space-time model.

Almost everybody has heard that time is the fourth dimension. Even though this concept is easy to imagine, people find it hard to understand because of its mystification by science popularizers. We live in three-dimensional space. This means that I can parameterize the universe such that I can describe any point in it by using just three letters (a, b, c).

Footnotes

1. All of these statements are only approximately true. They would be exactly true if the North Star was precisely overhead at the North Pole, instead of being off center by about 1.
2. The real estimate might not be 25,000 miles, as we do not know the exact correspondence between ancient and current measurements. However, this was a very good estimate for that time.
3. Despite the potential errors mentioned in footnote 2, this also was a good estimate for that time.