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These pyramids are related to geometry due to their shape, they have a square base and 4 slanted lines that meet at the top to form a point. These shapes are really important in geometry.

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The Moscow papyrus shows multiple algebraic problems. This includes calculating areas and volumes for surveying and construction.

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The Rhind papyrus is the foundation of mathematics, it includes various mathematical techniques like proportions, algebra, volume, and geometry.

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Section or Problem Numbers

Statement of Problem, or Description

Solution, or Description

Notes

Title Page

Ahmes identifies himself and his historical circumstances.

"Accurate reckoning. The entrance into the knowledge of all existing things and all obscure secrets. This book was copied in the year 33, in the fourth month of the inundation season, under the majesty of the king of Upper and Lower Egypt, 'A-user-Re', endowed with life, in likeness to writings of old made in the time of the king of Upper and Lower Egypt, Ne-ma'et-Re'. It is the scribe Ahmes who copies this writing."

It is clear from the title page that Ahmes identifies both his own period, as well as the period of an older text or texts from which he is supposed to have copied, thereby creating the Rhind Papyrus. The papyrus has material written on both sides-that is, its recto and verso. See the picture for details.

2/n Table

Express each of the quotients from 2/3 through 2/101 (where the denominator is always odd) as Egyptian fractions.

See the Rhind Mathematical Papyrus 2/n table article for summary and solutions of this section.

Throughout the papyrus, most solutions are given as particular Egyptian fractional representations of a given real number. However, since every positive rational number has infinitely many representations as an Egyptian fraction, these solutions are not unique. Also bear in mind that the fraction 2/3 is the single exception, used in addition to integers, that Ahmes uses alongside all (positive) rational unit fractions to express Egyptian fractions. The 2/n table can be said to partially follow an algorithm (see problem 61B) for expressing 2/n as an Egyptian fraction of 2 terms, when n is composite. However, this fledgling algorithm is cast aside in many situations when n is prime. The method of solutions for the 2/n table, therefore, also suggests beginnings of number theory, and not merely arithmetic.

1-9/10 Table

Write the quotients from 1/10 through 9/10 as Egyptian fractions.

Problems 1-6

1, 2, 6, 7, 8 and 9 loaves of bread (respectively, in each problem) are divided among 10 men. In each case, represent each man's share of bread as an Egyptian fraction.

The first six problems of the papyrus are simple repetitions of the information already written in the 1-9/10 table, now in the context of story problems.

7, 7B, 8-20

Let

and

.

Then for the following multiplications, write the product as an Egyptian fraction.

The same two multiplicands (here denoted as S and T) are used incessantly throughout these problems. Also note that Ahmes effectively writes the same problem thrice over (7, 7B, 10), sometimes approaching the same problem with different arithmetic work.

21-38

For each of the following linear equations with variable , solve for and express as an Egyptian fraction.

Note that problem 31 has an especially onerous solution. Although the statement of problems 21-38 can at times appear complicated (especially in Ahmes' prose), each problem ultimately reduces to a simple linear equation. In some cases, a unit of some kind has been omitted, being superfluous for these problems. These cases are problems 35-38, whose statements and "work" make the first mentions of units of volume known as a heqat and a ro (where 1 heqat = 320 ro), which will feature prominently throughout the rest of the papyrus. For the moment, however, their literal mention and usage in 35-38 is cosmetic.

39

100 bread loaves will be distributed unequally among 10 men. 50 loaves will be divided equally among 4 men so that each of those 4 receives an equal share , while the other 50 loaves will be divided equally among the other 6 men so that each of those 6 receives an equal share . Find the difference of these two shares and express same as an Egyptian fraction.

In problem 39, the papyrus begins to consider situations with more than one variable.

40

100 loaves of bread are to be divided among five men. The men's five shares of bread are to be in arithmetic progression, so that consecutive shares always differ by a fixed difference, or . Furthermore, the sum of the three largest shares is to be equal to seven times the sum of the two smallest shares. Find and write it as an Egyptian fraction.

Problem 40 concludes the arithmetic/algebraic section of the papyrus, to be followed by the geometry section. After problem 40, there is even a large section of blank space on the papyrus, which visually indicates the end of the section. As for problem 40 itself, Ahmes works out his solution by first considering the analogous case where the number of loaves is 60 as opposed to 100. He then states that in this case the difference is 5 1/2 and that the smallest share is equal to one, lists the others, and then scales his work back up to 100 to produce his result. Although Ahmes does not state the solution itself as it has been given here, the quantity is implicitly clear once he has re-scaled his first step by the multiplication 5/3 x 11/2, to list the five shares (which he does). It bears mentioning that this problem can be thought of as having four conditions: a) five shares sum to 100, b) the shares range from smallest to largest, c) consecutive shares have a constant difference and d) the sum of the three larger shares is equal to seven times the sum of the smaller two shares. Beginning with the first three conditions only, one can use elementary algebra and then consider whether adding the fourth condition yields a consistent result. It happens that once all four conditions are in place, the solution is unique. The problem is therefore a more elaborate case of linear equation solving than what has gone before, verging on linear algebra.

41

Use the volume formula

to calculate the volume of a cylindrical grain silo with a diameter of 9 cubits and a height of 10 cubits. Give the answer in terms of cubic cubits. Furthermore, given the following equalities among other units of volume, 1 cubic cubit = 3/2 khar = 30 heqats = 15/2 quadruple heqats, also express the answer in terms of khar and quadruple heqats.

This problem opens up the papyrus's geometry section, and also gives its first factually incorrect result (albeit with a very good approximation of , differing by less than one percent). Other ancient Egyptian volume units such as the quadruple heqat and the khar are later reported in this problem via unit conversion. Problem 41 is therefore also the first problem to treat significantly of dimensional analysis.

42

Reuse the volume formula and unit information given in 41 to calculate the volume of a cylindrical grain silo with a diameter of 10 cubits and a height of 10 cubits. Give the answer in terms of cubic cubits, khar, and hundreds of quadruple heqats, where 400 heqats = 100 quadruple heqats = 1 hundred-quadruple heqat, all as Egyptian fractions.

Problem 42 is effectively a repetition of 41, performing similar unit conversions at the end. However, although the problem does begin as stated, the arithmetic is considerably more involved, and certain of the given latter fractional terms are not actually present in the original document. However, the context is sufficient to fill in the gaps, and Chace has therefore taken license to add certain fractional terms in his mathematical translation (repeated here) which give rise to an internally consistent solution.

43

Use the volume formula

to calculate the volume of a cylindrical grain silo with a diameter of 9 cubits and a height of 6 cubits, directly finding the answer in Egyptian fractional terms of khar, and later in Egyptian fractional terms of quadruple heqats and quadruple ro, where 1 quadruple heqat = 4 heqat = 1280 ro = 320 quadruple ro.

Problem 43 represents the first serious mathematical mistake in the papyrus. Ahmes (or the source from which he may have been copying) attempted a shortcut in order to perform both the volume calculation and a unit conversion from cubic cubits to khar all in a single step, to avoid the need to use cubic cubits in an initial result. However, this attempt (which failed due to confusing part of the process used in 41 and 42 with that which was probably intended to be used in 43, giving consistent results by a different method) instead resulted in a new volume formula which is inconsistent with (and worse than) the approximation used in 41 and 42.

46

A rectangular prism-grain silo has a volume of 2500 quadruple heqats. Describe its three dimensions in terms of cubits.

This problem as stated has infinitely many solutions, but a simple choice of solution closely related to the terms of 44 and 45 is made.

47

Divide the physical volume quantity of 100 quadruple heqats by each of the multiples of 10, from 10 through 100. Express the results in Egyptian fractional terms of quadruple heqat and quadruple ro, and present the results in a table.

In problem 47, Ahmes is particularly insistent on representing more elaborate strings of fractions as Horus eye fractions, as far as he can. Compare problems 64 and 80 for similar preference of representation. To conserve space, "quadruple" has been shortened to "q." in all cases.

48

Compare the area of a circle with diameter 9 to that of its circumscribing square, which also has a side length of 9. What is the ratio of the area of the circle to that of the square?

The statement and solution of problem 48 make explicitly clear this preferred method of approximating the area of a circle, which had been used earlier in problems 41-43. However, it is erroneous. The original statement of problem 48 involves the usage of a unit of area known as the setat, which will shortly be given further context in future problems. For the moment, it is cosmetic.

51

A triangular tract of land has a base of 4 khet and an altitude of 10 khet. Find its area in terms of setat.

The setup and solution of 51 recall the familiar formula for calculating a triangle's area, and per Chace it is paraphrased as such. However, the papyrus's triangular diagram, previous mistakes, and translation issues present ambiguity over whether the triangle in question is a right triangle, or indeed if Ahmes actually understood the conditions under which the stated answer is correct. Specifically, it is unclear whether the dimension of 10 khet was meant as an altitude (in which case the problem is correctly worked as stated) or whether "10 khet" simply refers to a side of the triangle, in which case the figure would have to be a right triangle in order for the answer to be factually correct and properly worked, as done. These problems and confusions perpetuate themselves throughout 51-53, to the point where Ahmes seems to lose understanding of what he is doing, especially in 53.

53

Problem 53, being more complex, is fraught with many of the same issues as 51 and 52-translation ambiguities and several numerical mistakes. In particular concerning the large bottom trapezoid, Ahmes seems to get stuck on finding the upper base, and proposes in the original work to subtract "one tenth, equal to 1 + 1/4 + 1/8 setat plus 10 cubit strips" from a rectangle being (presumably) 4 1/2 x 3 1/2 (khet). However, even Ahmes' answer here is inconsistent with the problem's other information. Happily the context of 51 and 52, together with the base, mid-line, and smaller triangle area (which are given as 4 + 1/2, 2 + 1/4 and 7 + 1/2 + 1/4 + 1/8, respectively) make it possible to interpret the problem and its solution as has been done here. The given paraphrase therefore represents a consistent best guess as to the problem's intent, which follows Chace. Ahmes also refers to the "cubit strips" again in the course of calculating for this problem, and we therefore repeat their usage here. It bears mentioning that neither Ahmes nor Chace explicitly give the area for the middle trapezoid in their treatments (Chace suggests that this is a triviality from Ahmes' point of view); liberty has therefore been taken to report it in a manner which is consistent with what Chace had thus far advanced.

54

There are 10 plots of land. In each plot, a sector is partitioned off such that the sum of the area of these 10 new partitions is 7 setat. Each new partition has equal area. Find the area of any one of these 10 new partitions, and express it in Egyptian fractional terms of setat and cubit strips.

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55

There are 5 plots of land. In each plot, a sector is partitioned off such that the sum of the area of these 5 new partitions is 3 setat. Each new partition has equal area. Find the area of any one of these 5 new partitions, and express it in Egyptian fractional terms of setat and cubit strips.

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56

1) The unit of length known as a royal cubit is (and has been, throughout the papyrus) what is meant when we simply refer to a cubit. One royal cubit, or one cubit, is equal to seven palms, and one palm is equal to four fingers. In other words, the following equalities hold: 1 (royal) cubit = 1 cubit = 7 palms = 28 fingers.

2) Consider a right regular square pyramid whose base, the square face is coplanar with a plane (or the ground, say), so that any of the planes containing its triangular faces has the dihedral angle of with respect to the ground-plane (that is, on the interior of the pyramid). In other words, is the angle of the triangular faces of the pyramid with respect to the ground. The seked of such a pyramid, then, having altitude and base edge length , is defined as that physical length such that . Put another way, the seked of a pyramid can be interpreted as the ratio of its triangular faces' run per one unit (cubit) rise. Or, for the appropriate right triangle on a pyramid's interior having legs and the perpendicular bisector of a triangular face as the hypotenuse, then the pyramid's seked satisfies . Similar triangles are therefore described, and one can be scaled to the other.

3) A pyramid has an altitude of 250 (royal) cubits, and the side of its base has a length of 360 (royal) cubits. Find its seked in Egyptian fractional terms of (royal) cubits, and also in terms of palms.

Problem 56 is the first of the "pyramid problems" or seked problems in the Rhind papyrus, 56-59, 59B and 60, which concern the notion of a pyramid's facial inclination with respect to a flat ground. In this connection, the concept of a seked suggests early beginnings of trigonometry. Unlike modern trigonometry however, note especially that a seked is found with respect to some pyramid, and is itself a physical length measurement, which may be given in terms of any physical length units. For obvious reasons however, we (and the papyrus) confine our attention to situations involving ancient Egygtian units. We have also clarified that royal cubits are used throughout the papyrus, to differentiate them from "short" cubits which were used elsewhere in ancient Egypt. One "short" cubit is equal to six palms.

60

If a "pillar" (that is, a cone) has an altitude of 30 cubits, and the side of its base (or diameter) has a length of 15 cubits, find its seked and express it in terms of cubits.

Ahmes uses slightly different words to present this problem, which lend themselves to translation issues. However, the overall context of the problem, together with its accompanying diagram (which differs from the previous diagrams), leads Chace to conclude that a cone is meant. The notion of seked is easily generalized to the lateral face of a cone; he therefore reports the problem in these terms. Problem 60 concludes the geometry section of the papyrus. Moreover, it is the last problem on the recto (front side) of the document; all later content in this summary is present on the verso (back side) of the papyrus. The transition from 60 to 61 is thus both a thematic and physical shift in the papyrus.

61

Seventeen multiplications are to have their products expressed as Egyptian fractions. The whole is to be given as a table.

The syntax of the original document and its repeated multiplications indicate a rudimentary understanding that multiplication is commutative.

61B

Give a general procedure for converting the product of 2/3 and the reciprocal of any (positive) odd number 2n+1 into an Egyptian fraction of two terms, e.g. with natural p and q. In other words, find p and q in terms of n.

Problem 61B, and the method of decomposition that it describes (and suggests) is closely related to the computation of the Rhind Mathematical Papyrus 2/n table. In particular, every case in the 2/n table involving a denominator which is a multiple of 3 can be said to follow the example of 61B. 61B's statement and solution are also suggestive of a generality which most of the rest of the papyrus's more concrete problems do not have. It therefore represents an early suggestion of both algebra and algorithms.

63

700 loaves are to be divided unevenly among four men, in four unequal, weighted shares. The shares will be in the respective proportions . Find each share.

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64

Recall that the heqat is a unit of volume. Ten heqat of barley are to be distributed among ten men in an arithmetic progression, so that consecutive men's shares have a difference of 1/8 heqats. Find the ten shares and list them in descending order, in Egyptian fractional terms of heqat.

Problem 64 is a variant of 40, this time involving an even number of unknowns. For quick modern reference apart from Egyptian fractions, the shares range from 25/16 down through 7/16, where the numerator decreases by consecutive odd numbers. The terms are given as Horus eye fractions; compare problems 47 and 80 for more of this.

65

100 loaves of bread are to be unevenly divided among ten men. Seven of the men receive a single share, while the other three men, being a boatman, a foreman, and a door-keeper, each receive a double share. Express each of these two share amounts as Egyptian fractions.

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66

Recall that the heqat is a unit of volume and that one heqat equals 320 ro. 10 heqat of fat are distributed to one person over the course of one year (365 days), in daily allowances of equal amount. Express the allowance as an Egyptian fraction in terms of heqat and ro.

Problem 66 in its original form explicitly states that one year is equal to 365 days, and repeatedly uses the number 365 for its calculations. It is therefore primary historical evidence of the ancient Egyptian understanding of the year.

67

A shepherd had a flock of animals, and had to give a portion of his flock to a lord as tribute. The shepherd was told to give two-thirds OF one-third of his original flock as tribute. The shepherd gave 70 animals. Find the size of the shepherd's original flock.

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68

Four overseers are in charge of four crews of men, being 12, 8, 6 and 4 men, respectively. Each crewman works at a fungible rate, to produce a single work-product: production (picking, say) of grain. Working on some interval of time, these four gangs collectively produced 100 units, or 100 quadruple heqats of grain, where each crew's work-product will be given to each crew's overseer. Express each crew's output in terms of quadruple heqat.

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69

1) Consider cooking and food preparation. Suppose that there is a standardized way of cooking, or a production process, which will take volume units, specifically heqats of raw food-material (in particular, some one raw food-material) and produce units of some one finished food product. The pefsu of the (one) finished food product with respect to the (one) raw food-material, then, is defined as the quantity of finished food product units yielded from exactly one heqat of raw food material. In other words, .

2) 3 + 1/2 heqats of meal produce 80 loaves of bread. Find the meal per loaf in heqats and ro, and find the pefsu of these loaves with respect to the meal. Express them as Egyptian fractions.

Problem 69 begins the "pefsu" problems, 69-78, in the context of food preparation. Note that the notion of the pefsu assumes some standardized production process without accidents, waste, etc., and only concerns the relationship of one standardized finished food product to one particular raw material. That is, the pefsu is not immediately concerned with matters like production time, or (in any one given case) the relationship of other raw materials or equipment to the production process, etc. Still, the notion of the pefsu is another hint of abstraction in the papyrus, capable of being applied to any binary relationship between a food product (or finished good, for that matter) and a raw material. The concepts that the pefsu entails are thus typical of manufacturing.

71

1/2 heqats of besha, a raw material, produces exactly one full des-measure (glass) of beer. Suppose that there is a production process for diluted glasses of beer. 1/4 of the glass just described is poured out, and what has just been poured out is captured and re-used later. This glass, which is now 3/4 full, is then diluted back to capacity with water, producing exactly one full diluted glass of beer. Find the pefsu of these diluted beer glasses with respect to the besha as an Egyptian fraction.

Note that Problem 71 describes intermediate steps in a production process, as well as a second raw material, water. Further note that these are irrelevant to the relationship between the finished unit and the raw material (besha in this case).

72

Now that the concept of the pefsu has been established, problems 72-78 explore even exchanges of different heaps of finished foods, having different pefsu. In general however, they assume a common raw material of some kind. Specifically, the common raw material assumed throughout all of 72-78 is called wedyet flour, which is even implicated in the production of beer, so that beer can be exchanged for bread in the latter problems. 74's original statement also mentions "Upper Egyptian barley", but for our purposes this is cosmetic. What problems 72-78 say, then, is really this: equal amounts of raw material are used in two different production processes, to produce two different units of finished food, where each type has a different pefsu. One of the two finished food units is given. Find the other. This can be accomplished by dividing both units (known and unknown) by their respective pefsu, where the units of finished food vanish in dimensional analysis, and only the same raw material is considered. One can then easily solve for x. 72-78 therefore really require that x be given so that equal amounts of raw material are used in two different production processes.

74

1000 bread loaves of pefsu 5 are to be divided evenly into two heaps of 500 loaves each. Each heap is to be evenly exchanged for two other heaps, one of loaves of pefsu 10, and the other of loaves of pefsu 20. Find and .

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79

An estate's inventory consists of 7 houses, 49 cats, 343 mice, 2401 spelt plants (a type of wheat), and 16807 units of heqat (of whatever substance-a type of grain, suppose). List the items in the estates' inventory as a table, and include their total.

Problem 79 has been presented in its most literal interpretation. However, the problem is among the most interesting in the papyrus, as its setup and even method of solution suggests Geometric progression (that is, geometric sequences), elementary understanding of finite series, as well as the St. Ives problem-even Chace cannot help interrupting his own narrative in order to compare problem 79 with the St. Ives nursery rhyme. He also indicates that a suspiciously familiar third instance of these types of problems is to be found in Fibonacci's Liber Abaci. Chace suggests the interpretation that 79 is a kind of savings example, where a certain amount of grain is saved by keeping cats on hand to kill the mice which would otherwise eat the spelt used to make the grain. In the original document, the 2401 term is written as 2301 (an obvious mistake), while the other terms are given correctly; it is therefore corrected here.

Moreover, one of Ahmes' methods of solution for the sum suggests an understanding of finite geometric series. Ahmes performs a direct sum, but he also presents a simple multiplication to get the same answer: "2801 x 7 = 19607". Chace explains that since the first term, the number of houses (7) is equal to the common ratio of multiplication (7), then the following holds (and can be generalized to any similar situation):

That is, when the first term of a geometric sequence is equal to the common ratio, partial sums of geometric sequences, or finite geometric series, can be reduced to multiplications involving the finite series having one less term, which does prove convenient in this case. In this instance then, Ahmes simply adds the first four terms of the sequence (7 + 49 + 343 + 2401 = 2800) to produce a partial sum, adds one (2801), and then simply multiplies by 7 to produce the correct answer.

80

The hinu is a further unit of volume such that one heqat equals ten hinu. Consider the situations where one has a Horus eye fraction of heqats, and express their conversions to hinu in a table.

Compare problems 47 and 64 for other tabular information with repeated Horus eye fractions.

81

Perform "another reckoning of the hinu." That is, express an assortment of Egyptian fractions, many terms of which are also Horus eye fractions, in various terms of heqats, hinu, and ro.

Problem 81's main section is a much larger conversion table of assorted Egyptian fractions, which expands on the idea of problem 80-indeed, it represents one of the largest tabular forms in the entire papyrus. The first part of problem 81 is an exact repetition of the table in problem 80, without the first row which states that 1 heqat = 10 hinu; it is therefore not repeated here. The second part of problem 81, or its "body", is the large table which is given here. The attentive reader will notice two things: several rows repeat identical information, and several forms (but not all) given in both of the "heqat" areas on either side of the table are in fact identical. There are two points worth mentioning, to explain why the table looks the way that it does. For one thing, Ahmes does in fact exactly repeat certain groups of information in different areas of the table, and they are accordingly repeated here. On the other hand, Ahmes also starts out with certain "left-hand" heqat forms, and makes some mistakes in his early calculations. However, in many cases he corrects these mistakes later in his writing of the table, producing a consistent result. Since the present information is simply a re-creation of Chace's translation and interpretation of the papyrus, and since Chace elected to interpret and correct Ahmes' mistakes by substituting the later correct information in certain earlier rows, thereby fixing Ahmes' mistakes and also therefore repeating information in the course of translation, this method of interpretation explains the duplication of information in certain rows. As for the duplication of information in certain columns (1/4 heqat = ... = 1/4 heqat, etc.), this seems simply to have been a convention that Ahmes filled in while considering certain important Horus-eye fractional ratios from both the standpoint of the hinu, and also of the heqat (and their conversions). In short, the various repetitions of information are the result of choices made by Ahmes, his potential source document, and the editorial choices of Chace, in order to present a mathematically consistent translation of the larger table in problem 81.

82

Estimate in wedyet-flour, made into bread, the daily portion of feed for ten fattening geese. To do this, perform the following calculations, expressing the quantities in Egyptian fractional terms of hundreds of heqats, heqats and ro, except where specified otherwise:

Begin with the statement that "10 fattening geese eat 2 + 1/2 heqats in one day". In other words, the daily rate of consumption (and initial condition) is equal to 2 + 1/2. Determine the number of heqats which 10 fattening geese eat in 10 days, and in 40 days. Call these quantities and , respectively.

Multiply the above latter quantity by 5/3 to express the amount of "spelt", or , required to be ground up.

Multiply by 2/3 to express the amount of "wheat", or , required.

Divide by 10 to express a "portion of wheat", or , which is to be subtracted from .

Find . This is the amount of "grain", (or wedyet flour, it would seem), which is required to make the feed for geese, presumably on the interval of 40 days (which would seem to contradict the original statement of the problem, somewhat). Finally, express again in terms of hundreds of double heqats, double heqats and double ro, where 1 hundred double heqat = 2 hundred heqat = 100 double heqat = 200 heqat = 32000 double ro = 64000 ro. Call this final quantity .

Beginning with problem 82, the papyrus becomes increasingly difficult to interpret (owing to mistakes and missing information), to the point of unintelligibility. However, it is yet possible to make some sense of 82. Simply put, there seem to exist established rules, or good estimates, for fractions to be taken of this-or-that food material in a cooking or production process. Ahmes' 82 simply gives expression to some of these quantities, in what is after all declared in the original document to be an "estimate", its somewhat contradictory and confused language notwithstanding. In addition to their strangeness, problems 82, 82B, 83 and 84 are also notable for continuing the "food" train of thought of the recent pefsu problems, this time considering how to feed animals instead of people. Both 82 and 82B make use of the "hundred heqat" unit with regard to t and f; these conventions are cosmetic, and not repeated here. Licence is also taken throughout these last problems (per Chace) to fix numerical mistakes of the original document, to attempt to present a coherent paraphrase.

82B

Estimate the amount of feed for other geese. That is, consider a situation which is identical to problem 82, with the single exception that the initial condition, or daily rate of consumption, is exactly half as large. That is, let = 1 + 1/4. Find , and especially by using elementary algebra to skip the intermediate steps.

Problem 82B is presented in parallel with problem 82, and quickly considers the identical situation where the associated quantities are halved. In both cases, it appears that Ahmes' real goal is to find g_2. Now that he has a "procedure", he feels free to skip 82's onerous steps. One could simply observe that the division by two carries through the entire problem's work, so that g_2 is also exactly half as large as in problem 82. A slightly more thorough approach using elementary algebra would be to backtrack the relationships between the quantities in 82, make the essential observation that g = 14/15 x f, and then perform the unit conversions to transform g into g_2.

83

Estimate the feed for various kinds of birds. This is a "problem" with multiple components, which can be interpreted as a series of remarks:

Suppose that four geese are cooped up, and their collective daily allowance of feed is equal to one hinu. Express one goose's daily allowance of feed in terms of heqats and ro.

Suppose that the daily feed for a goose "that goes into the pond" is equal to 1/16 + 1/32 heqats + 2 ro. Express this same daily allowance in terms of hinu.

Suppose that the daily allowance of feed for 10 geese is one heqat. Find the 10-day allowance and the 30-day, or one-month allowance for the same group of animals, in heqats.

Finally a table will be presented, giving daily feed portions to fatten one animal of any of the indicated species.

Since problem 83's various items are concerned with unit conversions between heqats, ro and hinu, in the spirit of 80 and 81, it is natural to wonder what the table's items become when converted to hinu. The portion shared by the goose, terp-goose and crane is equal to 5/3 hinu, the set-ducks' portion is equal to 1/2 hinu, the ser-gooses' portion is equal to 1/4 hinu (compare the first item in the problem), and the portion shared by the dove and quail is equal to 1/16 + 1/32 hinu. The presence of various Horus eye fractions is familiar from the rest of the papyrus, and the table seems to consider feed estimates for birds, ranging from largest to smallest. The "5/3 hinu" portions at the top of the table, specifically its factor of 5/3, reminds one of the method for finding s in problem 82. Problem 83 makes mention of "Lower-Egyptian grain", or barley, and it also uses the "hundred-heqat" unit in one place; these are cosmetic, and left out of the present statement.

84

Estimate the feed for a stable of oxen.

84 is the last problem, or number, comprising the mathematical content of the Rhind papyrus. With regard to 84 itself, Chace echoes Peet: "One can only agree with Peet that 'with this problem the papyrus reaches its limit of unintelligibility and inaccuracy.'"(Chace, V.2, Problem 84). Here, instances of the "hundred heqat" unit have been expressed by "c. heqat" in order to conserve space. The three "cattle" mentioned are described as "common" cattle, to differentiate them from the other animals, and the two headers concerning loaves and "common food" are with respect to heqats. The "fine oxen" at the table's beginning are described as Upper Egyptian oxen, a phrase also removed here for space reasons.

Problem 84 seems to suggest a procedure to estimate various food materials and allowances in similar terms as the previous three problems, but the extant information is deeply confused. Still, there are hints of consistency. The problem seems to start out like a conventional story problem, describing a stable with ten animals of four different types. It seems that the four types of animals consume feed, or "loaves" at different rates, and that there are corresponding amounts of "common" food. These two columns of information are correctly summed in the "total" row, however they are followed by two "spelt" items of dubious relationship to the above. These two spelt items are indeed each multiplied by ten to give the two entries in the "10 days" row, once unit conversions are accounted for. The "one month" row items do not seem to be consistent with the previous two, however. Finally, information in "double heqats" (read hundred double heqats, double heqats and double ro for these items) concludes the problem, in a manner reminiscent of 82 and 82B. The two items in the final row are in roughly, but not exactly, the same proportion to one another as the two items in the "one month" row.

Number 85

A small group of cursive hieroglyphic signs is written, which Chace suggests may represent the scribe "trying his pen". It appears to be a phrase or sentence of some kind, and two translations are suggested. 1) "Kill vermin, mice, fresh weeds, numerous spiders. Pray the god Re for warmth, wind and high water." 2) Interpret this strange matter, which the scribe wrote ... according to what he knew."

The remaining items 85, 86 and 87, being various errata which are not mathematical in nature, are therefore styled by Chace as "numbers" as opposed to problems. They are also located on areas of the papyrus which are well away from the body of the writing, which had just ended with Problem 84. Number 85, for example, is some distance away from problem 84 on the verso-but not too far away. Its placement on the papyrus therefore suggests a kind of coda, in which case the latter translation, which Chace describes as an example of the "enigmatic writing" interpretation of ancient Egyptian documents, seems most appropriate to its context in the document.

Number 86

Number 86 seems to be from some account, or memorandum, and lists an assortment of goods and quantities, using words familiar from the context of the rest of the papyrus itself. [The original text is a series of lines of writing, which are therefore numbered in the following.]

"1... living forever. List of the food in Hebenti...

2... his brother the steward Ka-mose...

3... of his year, silver, 50 pieces twice in the year...

4... cattle 2, in silver 3 pieces in the year...

5... one twice; that is, 1/6 and 1/6. Now as for one...

6... 12 hinu; that is, silver, 1/4 piece; one...

7... (gold or silver) 5 pieces, their price therefor; fish, 120, twice...

8... year, barley, in quadruple heqat, 1/2 + 1/4 of 100 heqat 15 heqat; spelt, 100 heqat... heqat...

9... barley, in quadruple heqat, 1/2 + 1/4 of 100 heqat 15 heqat; spelt, 1 + 1/2 + 1/4 times 100 heqat 17 heqat...

10... 146 + 1/2; barley, 1 + 1/2 + 1/4 times 100 heqat 10 heqat; spelt, 300 heqat... heqat...

11... 1/2, there was brought wine, 1 ass(load?)...

12... silver 1/2 piece; ... 4; that is, in silver...

13... 1 + 1/4; fat, 36 hinu; that is, in silver...

14... 1 + 1/2 + 1/4 times 100 heqat 21 heqat; spelt, in quadruple heqat, 400 heqat 10 heqat...

15-18 (These lines are repetitions of line 14.)"

Chace indicates that number 86 was pasted onto the far left side of the verso (opposite the later geometry problems on the recto), to strengthen the papyrus. Number 86 can therefore be interpreted as a piece of "scrap paper".

Number 87

Number 87 is a brief account of certain events. Chace indicates an (admittedly now dated and possibly changed) scholarly consensus that 87 was added to the papyrus not long after the completion of its mathematical content. He goes on to indicate that the events described in it "took place during the period of the Hyksos domination."

"Year 11, second month of the harvest season. Heliopolis was entered.

The first month of the inundation season, 23rd day, the commander (?) of the army (?) attacked (?) Zaru.

25th day, it was heard that Zaru was entered.

Year 11, first month of the inundation season, third day. Birth of Set; the majesty of this god caused his voice to be heard.

Birth of Isis, the heavens rained."

Number 87 is located toward the middle of the verso, surrounded by a large, blank, unused space.

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Section or Problem Numbers Statement of Problem, or Description Solution, or Description Notes Title Page Ahmes identifies himself and his historical circumstances. "Accurate reckoning. The entrance into the knowledge of all existing things and all obscure secrets.

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This showcased geometric principals in its agriculture around 447-432 BC.

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Discovered by Pythagoras and his followers in the 6th century BC.

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Laid the foundation for systematic study and understanding of geometric principals.

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Established a rigorous method for verifying geometric propositions.

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Islamic agriculture during the golden age of Islam showcased intricate geometric patterns and symmetry.

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This expanded the understanding of perspective and spatial relationships.

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Artists used geometric principals to create realistic perspectives and proportions in their painting. Also in their sculptures.

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Allowed the representation of geometric shapes using algebraic equations, using mathematics and enabling the development of calculus.

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Architects used geometric forms to create innovate and functional building designs that departed from traditional architectural styles.

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This revealed complex geometric shapes that are similar at different scales.

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In mathematics, a fractal is a self-similar subset of Euclidean space whose fractal dimension strictly exceeds its topological dimension. Fractals appear the same at different levels, as illustrated in successive magnifications of the Mandelbrot set. Fractals exhibit similar patterns at increasingly small scales called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, it is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.

One way that fractals are different from finite geometric figures is the way in which they scale. Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension.

Analytically, fractals are usually nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line - although it is still 1-dimensional, its fractal dimension indicates that it also resembles a surface.

Starting in the 17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass, and on to the coining of the word in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century. The term "fractal" was first used by mathematician Benoit Mandelbrot in 1975. Mandelbrot based it on the Latin , meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.

There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals." More formally, in 1982 Mandelbrot stated that "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Later, seeing this as too restrictive, he simplified and expanded the definition to: "A fractal is a shape made of parts similar to the whole in some way." Still later, Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants".

The consensus is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth. Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology, art, architecture and law. Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractals.

The word "fractal" often has different connotations for the lay public as opposed to mathematicians, where the public is more likely to be familiar with fractal art than the mathematical concept. The mathematical concept is difficult to define formally, even for mathematicians, but key features can be understood with a little mathematical background.

The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. Self-similarity itself is not necessarily counter-intuitive (e.g., people have pondered self-similarity informally such as in the infinite regress in parallel mirrors or the homunculus, the little man inside the head of the little man inside the head ...). The difference for fractals is that the pattern reproduced must be detailed.

This idea of being detailed relates to another feature that can be understood without much mathematical background: Having a fractal dimension greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric shapes are usually perceived. A straight line, for instance, is conventionally understood to be one-dimensional; if such a figure is rep-tiled into pieces each 1/3 the length of the original, then there are always three equal pieces. A solid square is understood to be two-dimensional; if such a figure is rep-tiled pieces each scaled down by a factor of 1/3 in both dimensions, there are a total of 3 ² = 9 pieces. We see that for ordinary self-similar objects, being n-dimensional means that when it is rep-tiled into pieces each scaled down by a scale-factor of 1/ r, there are a total of r n pieces. Now, consider the Koch curve. It can be rep-tiled into four sub-copies, each scaled down by a scale-factor of 1/3. So, strictly by analogy, we can consider the "dimension" of the Koch curve as being the unique real number D that satisfies 3D = 4. This number is what mathematicians call the fractal dimension of the Koch curve; it is certainly not what is conventionally perceived as the dimension of a curve (this number is not even an integer!). The fact that the Koch curve has a fractal dimension differing from its conventionally understood dimension (that is, its topological dimension) is what makes it a fractal.

This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere differentiable". In a concrete sense, this means fractals cannot be measured in traditional ways. To elaborate, in trying to find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of measuring with a tape measure. But in measuring an infinitely "wiggly" fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the jagged pattern would always re-appear, at arbitrarily small scales, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve. The result is that one must need infinite tape to perfectly cover the entire curve, i.e. the snowflake has an infinite perimeter.

The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, with several notable people contributing canonical fractal forms along the way. A common theme in ancient traditional African architecture is the use of fractal scaling, whereby small parts of the structure tend to look similar to larger parts, such as a circular village made of circular houses. According to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense). In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them. Indeed, according to various historical accounts, after that point few mathematicians tackled the issues and the work of those who did remained obscured largely because of resistance to such unfamiliar emerging concepts, which were sometimes referred to as mathematical "monsters". Thus, it was not until two centuries had passed that on July 18, 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered a fractal, having the non- intuitive property of being everywhere continuous but nowhere differentiable at the Royal Prussian Academy of Sciences. In addition, the quotient difference becomes arbitrarily large as the summation index increases. Not long after that, in 1883, Georg Cantor, who attended lectures by Weierstrass, published examples of subsets of the real line known as Cantor sets, which had unusual properties and are now recognized as fractals. Also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called "self-inverse" fractals.

One of the next milestones came in 1904, when Helge von Koch, extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand-drawn images of a similar function, which is now called the Koch snowflake. Another milestone came a decade later in 1915, when Wacław Sierpiński constructed his famous triangle then, one year later, his carpet. By 1918, two French mathematicians, Pierre Fatou and Gaston Julia, though working independently, arrived essentially simultaneously at results describing what is now seen as fractal behaviour associated with mapping complex numbers and iterative functions and leading to further ideas about attractors and repellors (i.e., points that attract or repel other points), which have become very important in the study of fractals. Very shortly after that work was submitted, by March 1918, Felix Hausdorff expanded the definition of "dimension", significantly for the evolution of the definition of fractals, to allow for sets to have non-integer dimensions. The idea of self-similar curves was taken further by Paul Lévy, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole, described a new fractal curve, the Lévy C curve.

Different researchers have postulated that without the aid of modern computer graphics, early investigators were limited to what they could depict in manual drawings, so lacked the means to visualize the beauty and appreciate some of the implications of many of the patterns they had discovered (the Julia set, for instance, could only be visualized through a few iterations as very simple drawings). That changed, however, in the 1960s, when Benoit Mandelbrot started writing about self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, which built on earlier work by Lewis Fry Richardson. In 1975 Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word "fractal" and illustrated his mathematical definition with striking computer-constructed visualizations. These images, such as of his canonical Mandelbrot set, captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".

In 1980, Loren Carpenter gave a presentation at the SIGGRAPH where he introduced his software for generating and rendering fractally generated landscapes.

One often cited description that Mandelbrot published to describe geometric fractals is "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole"; this is generally helpful but limited. Authors disagree on the exact definition of fractal, but most usually elaborate on the basic ideas of self-similarity and the unusual relationship fractals have with the space they are embedded in.

One point agreed on is that fractal patterns are characterized by fractal dimensions, but whereas these numbers quantify complexity (i.e., changing detail with changing scale), they neither uniquely describe nor specify details of how to construct particular fractal patterns. In 1975 when Mandelbrot coined the word "fractal", he did so to denote an object whose Hausdorff-Besicovitch dimension is greater than its topological dimension. However, this requirement is not met by space-filling curves such as the Hilbert curve.

Because of the trouble involved in finding one definition for fractals, some argue that fractals should not be strictly defined at all. According to Falconer, fractals should, in addition to being nowhere differentiable and able to have a fractal dimension, be only generally characterized by a gestalt of the following features;

Self-similarity, which may include:

Exact self-similarity: identical at all scales, such as the Koch snowflake

Quasi self-similarity: approximates the same pattern at different scales; may contain small copies of the entire fractal in distorted and degenerate forms; e.g., the Mandelbrot set's satellites are approximations of the entire set, but not exact copies.

Statistical self-similarity: repeats a pattern stochastically so numerical or statistical measures are preserved across scales; e.g., randomly generated fractals like the well-known example of the coastline of Britain for which one would not expect to find a segment scaled and repeated as neatly as the repeated unit that defines fractals like the Koch snowflake.

Qualitative self-similarity: as in a time series

Multifractal scaling: characterized by more than one fractal dimension or scaling rule

Fine or detailed structure at arbitrarily small scales. A consequence of this structure is fractals may have emergent properties (related to the next criterion in this list).

Irregularity locally and globally that is not easily described in traditional Euclidean geometric language. For images of fractal patterns, this has been expressed by phrases such as "smoothly piling up surfaces" and "swirls upon swirls".

Simple and "perhaps recursive" definitions; see Common techniques for generating fractals

As a group, these criteria form guidelines for excluding certain cases, such as those that may be self-similar without having other typically fractal features. A straight line, for instance, is self-similar but not fractal because it lacks detail, is easily described in Euclidean language, has the same Hausdorff dimension as topological dimension, and is fully defined without a need for recursion.

Images of fractals can be created by fractal generating programs. Because of the butterfly effect, a small change in a single variable can have an unpredictable outcome.

Iterated function systems (IFS) - use fixed geometric replacement rules; may be stochastic or deterministic; e.g., Koch snowflake, Cantor set, Haferman carpet, Sierpinski carpet, Sierpinski gasket, Peano curve, Harter-Heighway dragon curve, T-square, Menger sponge

Strange attractors - use iterations of a map or solutions of a system of initial-value differential or difference equations that exhibit chaos (e.g., see multifractal image, or the logistic map)

- use string rewriting; may resemble branching patterns, such as in plants, biological cells (e.g., neurons and immune system cells), blood vessels, pulmonary structure, etc. or turtle graphics patterns such as space-filling curves and tilings

Escape-time fractals - use a formula or recurrence relation at each point in a space (such as the complex plane); usually quasi-self-similar; also known as "orbit" fractals; e.g., the Mandelbrot set, Julia set, Burning Ship fractal, Nova fractal and Lyapunov fractal. The 2d vector fields that are generated by one or two iterations of escape-time formulae also give rise to a fractal form when points (or pixel data) are passed through this field repeatedly.

Random fractals - use stochastic rules; e.g., Lévy flight, percolation clusters, self avoiding walks, fractal landscapes, trajectories of Brownian motion and the Brownian tree (i.e., dendritic fractals generated by modeling diffusion-limited aggregation or reaction-limited aggregation clusters).

Fractal patterns have been modeled extensively, albeit within a range of scales rather than infinitely, owing to the practical limits of physical time and space. Models may simulate theoretical fractals or natural phenomena with fractal features. The outputs of the modelling process may be highly artistic renderings, outputs for investigation, or benchmarks for fractal analysis. Some specific applications of fractals to technology are listed elsewhere. Images and other outputs of modelling are normally referred to as being "fractals" even if they do not have strictly fractal characteristics, such as when it is possible to zoom into a region of the fractal image that does not exhibit any fractal properties. Also, these may include calculation or display artifacts which are not characteristics of true fractals.

Modeled fractals may be sounds, digital images, electrochemical patterns, circadian rhythms, etc. Fractal patterns have been reconstructed in physical 3-dimensional space and virtually, often called " in silico" modeling. Models of fractals are generally created using fractal-generating software that implements techniques such as those outlined above. As one illustration, trees, ferns, cells of the nervous system, blood and lung vasculature, and other branching patterns in nature can be modeled on a computer by using recursive algorithms and L-systems techniques. The recursive nature of some patterns is obvious in certain examples-a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. Similarly, random fractals have been used to describe/create many highly irregular real-world objects. A limitation of modeling fractals is that resemblance of a fractal model to a natural phenomenon does not prove that the phenomenon being modeled is formed by a process similar to the modeling algorithms.

Approximate fractals found in nature display self-similarity over extended, but finite, scale ranges. The connection between fractals and leaves, for instance, is currently being used to determine how much carbon is contained in trees. Phenomena known to have fractal features include:

A fractal is formed when pulling apart two glue-covered acrylic sheets

High-voltage breakdown within a 4 in (100 mm) block of acrylic glass creates a fractal Lichtenberg figure

Fractal defrosting patterns, polar Mars. The patterns are formed by sublimation of frozen CO ₂. Width of image is about a kilometer.

Since 1999, more than 10 scientific groups have performed fractal analysis on over 50 of Jackson Pollock's (1912-1956) paintings which were created by pouring paint directly onto his horizontal canvases Recently, fractal analysis has been used to achieve a 93% success rate in distinguishing real from imitation Pollocks. Cognitive neuroscientists have shown that Pollock's fractals induce the same stress-reduction in observers as computer-generated fractals and Nature's fractals.

Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns. It involves pressing paint between two surfaces and pulling them apart.

Cyberneticist Ron Eglash has suggested that fractal geometry and mathematics are prevalent in African art, games, divination, trade, and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles. Hokky Situngkir also suggested the similar properties in Indonesian traditional art, batik, and ornaments found in traditional houses.

Ethnomathematician Ron Eglash has discussed the planned layout of Benin city using fractals as the basis, not only in the city itself and the villages but even in the rooms of houses. He commented that "When Europeans first came to Africa, they considered the architecture very disorganised and thus primitive. It never occurred to them that the Africans might have been using a form of mathematics that they hadn't even discovered yet."

In a 1996 interview with Michael Silverblatt, David Foster Wallace admitted that the structure of the first draft of Infinite Jest he gave to his editor Michael Pietsch was inspired by fractals, specifically the Sierpinski triangle (a.k.a. Sierpinski gasket), but that the edited novel is "more like a lopsided Sierpinsky Gasket".

Some works by the Dutch artist M. C. Escher, such as Circle Limit III, contain shapes repeated to infinity that become smaller and smaller as they get near to the edges, in a pattern that would always look the same if zoomed in.

Humans appear to be especially well-adapted to processing fractal patterns with D values between 1.3 and 1.5. When humans view fractal patterns with D values between 1.3 and 1.5, this tends to reduce physiological stress.

Ion propulsion

When two-dimensional fractals are iterated many times, the perimeter of the fractal increases up to infinity, but the area may never exceed a certain value. A fractal in three-dimensional space is similar; such a fractal may have an infinite surface area, but never exceed a certain volume. This can be utilized to maximize the efficiency of ion propulsion when choosing electron emitter construction and material. If done correctly, the efficiency of the emission process can be maximized.

Jürgens, Hartmut; Peitgen, Heinz-Otto; and Saupe, Dietmar; Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992. ISBN 0-387-97903-4

Mandelbrot, Benoit B.; The Fractal Geometry of Nature. New York: W. H. Freeman and Co., 1982. ISBN 0-7167-1186-9

Pickover, Clifford A.; ed.; Chaos and Fractals: A Computer Graphical Journey - A 10 Year Compilation of Advanced Research. Elsevier, 1998. ISBN 0-444-50002-2

Lauwerier, Hans; Fractals: Endlessly Repeated Geometrical Figures, Translated by Sophia Gill-Hoffstadt, Princeton University Press, Princeton NJ, 1991. ISBN 0-691-08551-X, cloth. ISBN 0-691-02445-6 paperback. "This book has been written for a wide audience..." Includes sample BASIC programs in an appendix.

Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 978-0-19-850839-7.

Wahl, Bernt; Van Roy, Peter; Larsen, Michael; and Kampman, Eric; Exploring Fractals on the Macintosh, Addison Wesley, 1995. ISBN 0-201-62630-6

Lesmoir-Gordon, Nigel; The Colours of Infinity: The Beauty, The Power and the Sense of Fractals. 2004. ISBN 1-904555-05-5 (The book comes with a related DVD of the Arthur C. Clarke documentary introduction to the fractal concept and the Mandelbrot set.)

Gouyet, Jean-François; Physics and Fractal Structures (Foreword by B. Mandelbrot); Masson, 1996. ISBN 2-225-85130-1, and New York: Springer-Verlag, 1996. ISBN 978-0-387-94153-0. Out-of-print. Available in PDF version at."Physics and Fractal Structures" (in French). Jfgouyet.fr.

Bunde, Armin; Havlin, Shlomo (1996). Fractals and Disordered Systems. Springer.

Bunde, Armin; Havlin, Shlomo (1995). Fractals in Science. Springer.

ben-Avraham, Daniel; Havlin, Shlomo (2000). Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press.

Falconer, Kenneth (2013). Fractals, A Very Short Introduction. Oxford University Press.

^ Santo Banerjee, M. K. Hassan, Sayan Mukherjee and A. Gowrisankar, Fractal Patterns in Nonlinear Dynamics and Applications. (CRC Press, Taylor & Francis Group, 2019)

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In mathematics, a fractal is a self-similar subset of Euclidean space whose fractal dimension strictly exceeds its topological dimension. Fractals appear the same at different levels, as illustrated in successive magnifications of the Mandelbrot set.

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Mathmaticians challenged the traditional Euclidean postulates. Lead to the new understandings of space and laying the groundwork for Einsteins theory.

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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry.

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The layout of Stonehenge shows an understanding of geometric principles, in its circular arrangement and alignment with astronomical events like solstices.

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Made numerous contributions to geometry, the calculation of areas and volumes of various shapes.

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The Romans used geometric principals to construct aqueducts, bridges, buildings etc.

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Studies properties of space that are preserved under continuous deformations.

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States that any map can be colored with no more than 4 colors.

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The development of algorithms for rendering 3D objects and animations relies heavily on geometric principles.