DOCSLIB.ORG

High Precision Division and Square Root

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
High Precision Division and Square Root High Precision Division and Square Ro ot Alan H. Karp and Peter Markstein Hewlett-Packard Lab oratories 1501 Page Mill Road Palo Alto, CA 94304 We present division and square ro ot algorithms for calculations with more bits than are handled by the oating p oint hardware. These algorithms avoid the need to multiply two high precision numb ers, sp eeding up the last iteration byasmuch as a factor of ten. Wealsoshowhow to pro duce the oating p ointnumb er closest to the exact result with relatively few additional op erations. Categories and Sub ject Descriptors: G.1.0 [Numerical Analysis]: Computer Arithmetic; G.4 [Mathematical Software]: Algorithm Analysis General Terms: Floating p oint arithmetic Additional Key Words and Phrases: Quad precision, square ro ot, division 1. INTRODUCTION Floating p oint division and square ro ot take considerably longer to compute than addition and multiplication. The latter two are computed directly while the former are usually computed with an iterative algorithm. The most common approach is to use a division-free Newton-Raphson iteration to get an approximation to the recipro cal of the denominator (division) or the recipro cal square ro ot, and then multiply bythenumerator (division) or input argument (square ro ot). Done care- fully, this approach can return a oating p ointnumb er within 1 or 2 ULPs (Units in the Last Place) of the oating p ointnumb er closest to the exact result. With more work, sometimes a lot more, the error can b e reduced to half an ULP or less. Name: Alan H. Karp Aliation: Future Systems Department Address: alan [email protected] Name: Peter Markstein Aliation: Computer Systems Lab oratory [email protected] Address: p eter Permission to make digital or hard copies of part or all of this work for p ersonal or classro om use is granted without fee provided that copies are not made or distributed for pro t or direct commercial advantage and that copies show this notice on the rst page or initial screen of a display along with the full citation. Copyrights for comp onents of this work owned by others than ACM must b e honored. Abstracting with credit is p ermitted. To copy otherwise, to republish, to p ost on servers, to redistribute to lists, or to use any comp onentofthiswork in other works, requires prior sp eci c p ermission and/or a fee. Permissions may b e requested from Publications Dept, ACM Inc., 1515 Broadway, New York, NY 10036 USA, fax +1 (212) 869-0481, or [email protected]. 2 Alan H. Karp and Peter Markstein The usual approachworks quite well when the results are computed to no more precision than the hardware addition and multiplication. Typically, division and square ro ot take10to20machine cycles on a pro cessor that do es a multiplication in the same precision in 2 or 3 cycles. The situation is not so go o d for higher precision results b ecause the cost of the nal multiplication is much higher. The key feature of our algorithm is that it avoids this exp ensivemultiplication. Section 2 discusses some metho ds used to do division and square ro ot. Section 3 describ es the oating p oint arithmetic and the assumptions we are making ab out the oating p oint hardware. Section 4 describ es our new algorithms. In Section 5 we present an analysis of the accuracy of the algorithm. Next, weshowhowto get the correctly rounded results with relatively few additional op erations, on av- erage, in Section 6. The pro cedures we use for testing are describ ed in Section 7. Programs written in the programming language of the Unix desk top calculator utility bc[Hewlett-Packard 1991] that implement these algorithms are included in an app endix. 2. ALGORITHMIC CHOICES There are a number of ways to p erform division and square ro ot. In this section we will discuss these metho ds with emphasis on results that exceed the precision of the oating p oint hardware. One approach istodoscho ol-b oy long division, a scheme identical to successive subtraction for base 2 arithmetic[Goldb erg 1990]. The pro cedure is straightforward. To compute B=A,we initialize a remainder to R =0.LetS b e a shift register containing R concatenated with B . Then, for each bit we do the following steps: (1) Shift S left one bit so that the high order bit of B b ecomes the low order bit of R. (2) Subtract A from from the R part of S . (3) If the result is negative, set the low order bit of the B part of S to zero; otherwise set it to 1. (4) If the result of the subtraction is negative, add A to the R part of S . At the end of N steps, the N-bit quotientisinB , and the remainder is in R. A similar scheme can b e used for square ro ot[Monuschi and Mezzalama 1990]. We start with the input value A and a rst guess Y =0.For each bit we compute 2 the residual, R = A Y . Insp ection of R at the k 'th step is used to select a value for the k 'th bit of Y . The selection rules give some exibilityintuningthe algorithm. There are improvements to these approaches. For example, non-restoring division avoids adding A to R.TheSRT algorithm[Patterson and Hennessy 1990] is an opti- mization based on a carry-sum intermediate representation. Higher radix metho ds compute several bits p er step. Unfortunately, these improvements can't overcome the drawbackthatwe only get a xed numb er of bits p er step, a particularly serious shortcoming for high precision computations. Polynomialapproximations can also b e used[Monuschi and Mezzalama 1990]. Chebyshev p olynomials are most often used b ecause they form an approximation with a guaranteed b ound on the maximum error. However, each term in the series High Precision Division and Square Ro ot 3 adds ab out the same numb er of bits to the precision of the result, so this approach is impractical for high precision results. Another approach is one of the CORDIC metho ds whichwere originally derived for trigonometric functions but can b e applied to square ro ot[Monuschi and Mez- zalama 1990]. These metho ds treat the input as a vector in a particular co ordinate system. The result is then computed by adding and shifting some tabulated num- bers. However, the tables would havetobeinconveniently large for high precision results. There are two metho ds in common use that converge quadratically instead of linearly as do these other metho ds, Goldschmidt's metho d and Newton iteration. Quadratic convergence is particularly imp ortant for high precision calculations b e- cause the numb er of steps is prop ortional to the logarithm of the numberofdigits. Goldschmidt's algorithm[Patterson and Hennessy 1990] is based on a dual iter- 0 ation. TocomputeB=A we rst nd an approximationto1=A = A .Wethen 0 0 initialize x = BA and y = AA .Next,we iterate until convergence. (1) r =2 y . (2) y = ry. (3) x = rx. 0 If 0 AA < 2, the iteration converges quadratically. A similar algorithm can b e derived for square ro ot. Round-o errors will not b e damp ed out b ecause Goldschmidt's algorithm is not self-correcting. In order to get accurate results, we will havetomaintain throughout the calculation more digits than app ear in the nal result. While carrying an extra word of precision is not a problem in a multiprecision calculation, the extra digits required will slowdown a calculation that could have b een done with quad precision hardware. Newton's metho d for computing B=A is to approximate 1=A, apply several itera- tions of the Newton-Raphson metho d, and multiply the result by B . The iteration for the recipro cal of A is x = x + x (1 Ax ): (1) n+1 n n n p p Newton's metho d for computing A is to approximate 1= A, apply several iterations of the Newton-Raphson metho d, and multiply the result by A. The iteration for the recipro cal square ro ot of A is x n 2 x = x + (1 Ax ): (2) n+1 n n 2 Newton's metho d is quadratically convergent and self-correcting. Thus, round- o errors made in early iterations damp out. In particular, if we know that the rst k guess is accurate to N bits, the result of iteration k will b e accurate to almost 2 N bits. 3. FLOATING POINT ARITHMETIC Our mo di cation to the standard Newton metho d for division and square ro ot makes some assumptions ab out the oating p oint arithmetic. In this section we intro duce some terminology, describ e the hardware requirements to implementour algorithm, and showhow to use software to overcome any de ciencies. 4 Alan H. Karp and Peter Markstein We assume that wehave a oating p oint arithmetic that allows us to write any representable number N X e k x = f ; (3) k k =0 where is the numb er base, the integer e is the exp onent, and the integers f are k in the interval [0; ). As will b e seen in Section 4, we compute a 2N -bit approximation to our function from an N -bit approximation. We will refer to these as full precision and half precision numb ers, resp ectively. In the sp ecial case where the op erations on half precision numb ers are done in hardware, we will refer to 2N -bit numb ers as quad precision and N -bit numb ers as hardware precision.
Load more

Recommended publications
  • Lesson 19: the Euclidean Algorithm As an Application of the Long Division Algorithm
    NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 6•2 Lesson 19: The Euclidean Algorithm as an Application of the Long Division Algorithm Student Outcomes . Students explore and discover that Euclid’s Algorithm is a more efficient means to finding the greatest common factor of larger numbers and determine that Euclid’s Algorithm is based on long division. Lesson Notes MP.7 Students look for and make use of structure, connecting long division to Euclid’s Algorithm. Students look for and express regularity in repeated calculations leading to finding the greatest common factor of a pair of numbers. These steps are contained in the Student Materials and should be reproduced, so they can be displayed throughout the lesson: Euclid’s Algorithm is used to find the greatest common factor (GCF) of two whole numbers. MP.8 1. Divide the larger of the two numbers by the smaller one. 2. If there is a remainder, divide it into the divisor. 3. Continue dividing the last divisor by the last remainder until the remainder is zero. 4. The final divisor is the GCF of the original pair of numbers. In application, the algorithm can be used to find the side length of the largest square that can be used to completely fill a rectangle so that there is no overlap or gaps. Classwork Opening (5 minutes) Lesson 18 Problem Set can be discussed before going on to this lesson. Lesson 19: The Euclidean Algorithm as an Application of the Long Division Algorithm 178 Date: 4/1/14 This work is licensed under a © 2013 Common Core, Inc.
    [Show full text]
  • Primality Testing for Beginners
    STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker http://dx.doi.org/10.1090/stml/070 Primality Testing for Beginners STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell Gerald B. Folland (Chair) Serge Tabachnikov The cover illustration is a variant of the Sieve of Eratosthenes (Sec- tion 1.5), showing the integers from 1 to 2704 colored by the number of their prime factors, including repeats. The illustration was created us- ing MATLAB. The back cover shows a phase plot of the Riemann zeta function (see Appendix A), which appears courtesy of Elias Wegert (www.visual.wegert.com). 2010 Mathematics Subject Classification. Primary 11-01, 11-02, 11Axx, 11Y11, 11Y16. For additional information and updates on this book, visit www.ams.org/bookpages/stml-70 Library of Congress Cataloging-in-Publication Data Rempe-Gillen, Lasse, 1978– author. [Primzahltests f¨ur Einsteiger. English] Primality testing for beginners / Lasse Rempe-Gillen, Rebecca Waldecker. pages cm. — (Student mathematical library ; volume 70) Translation of: Primzahltests f¨ur Einsteiger : Zahlentheorie - Algorithmik - Kryptographie. Includes bibliographical references and index. ISBN 978-0-8218-9883-3 (alk. paper) 1. Number theory. I. Waldecker, Rebecca, 1979– author. II. Title. QA241.R45813 2014 512.72—dc23 2013032423 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.
    [Show full text]
  • So You Think You Can Divide?
    So You Think You Can Divide? A History of Division Stephen Lucas Department of Mathematics and Statistics James Madison University, Harrisonburg VA October 10, 2011 Tobias Dantzig: Number (1930, p26) “There is a story of a German merchant of the fifteenth century, which I have not succeeded in authenticating, but it is so characteristic of the situation then existing that I cannot resist the temptation of telling it. It appears that the merchant had a son whom he desired to give an advanced commercial education. He appealed to a prominent professor of a university for advice as to where he should send his son. The reply was that if the mathematical curriculum of the young man was to be confined to adding and subtracting, he perhaps could obtain the instruction in a German university; but the art of multiplying and dividing, he continued, had been greatly developed in Italy, which in his opinion was the only country where such advanced instruction could be obtained.” Ancient Techniques Positional Notation Division Yielding Decimals Outline Ancient Techniques Division Yielding Decimals Definitions Integer Division Successive Subtraction Modern Division Doubling Multiply by Reciprocal Geometry Iteration – Newton Positional Notation Iteration – Goldschmidt Iteration – EDSAC Positional Definition Galley or Scratch Factor Napier’s Rods and the “Modern” method Short Division and Genaille’s Rods Double Division Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Definitions If a and b are natural numbers and a = qb + r, where q is a nonnegative integer and r is an integer satisfying 0 ≤ r < b, then q is the quotient and r is the remainder after integer division.
    [Show full text]
  • Decimal Long Division EM3TLG1 G5 466Z-NEW.Qx 6/20/08 11:42 AM Page 557
    EM3TLG1_G5_466Z-NEW.qx 6/20/08 11:42 AM Page 556 JE PRO CT Objective To extend the long division algorithm to problems in which both the divisor and the dividend are decimals. 1 Doing the Project materials Recommended Use During or after Lesson 4-6 and Project 5. ٗ Math Journal, p. 16 ,Key Activities ٗ Student Reference Book Students explore the meaning of division by a decimal and extend long division to pp. 37, 54G, 54H, and 60 decimal divisors. Key Concepts and Skills • Use long division to solve division problems with decimal divisors. [Operations and Computation Goal 3] • Multiply numbers by powers of 10. [Operations and Computation Goal 3] • Use the Multiplication Rule to find equivalent fractions. [Number and Numeration Goal 5] • Explore the meaning of division by a decimal. [Operations and Computation Goal 7] Key Vocabulary decimal divisors • dividend • divisor 2 Extending the Project materials Students express the remainder in a division problem as a whole number, a fraction, an ٗ Math Journal, p. 17 exact decimal, and a decimal rounded to the nearest hundredth. ٗ Student Reference Book, p. 54I Technology See the iTLG. 466Z Project 14 Decimal Long Division EM3TLG1_G5_466Z-NEW.qx 6/20/08 11:42 AM Page 557 Student Page 1 Doing the Project Date PROJECT 14 Dividing with Decimal Divisors WHOLE-CLASS 1. Draw lines to connect each number model with the number story that fits it best. Number Model Number Story ▼ Exploring Meanings for DISCUSSION What is the area of a rectangle 1.75 m by ?cm 50 0.10 ء Decimal Division 2 Sales tax is 10%.
    [Show full text]
  • Basics of Math 1 Logic and Sets the Statement a Is True, B Is False
    Basics of math 1 Logic and sets The statement a is true, b is false. Both statements are false. 9000086601 (level 1): Let a and b be two sentences in the sense of mathematical logic. It is known that the composite statement 9000086604 (level 1): Let a and b be two sentences in the sense of mathematical :(a _ b) logic. It is known that the composite statement is true. For each a and b determine whether it is true or false. :(a ^ :b) is false. For each a and b determine whether it is true or false. Both statements are false. The statement a is true, b is false. Both statements are true. The statement a is true, b is false. Both statements are true. The statement a is false, b is true. The statement a is false, b is true. Both statements are false. 9000086602 (level 1): Let a and b be two sentences in the sense of mathematical logic. It is known that the composite statement 9000086605 (level 1): Let a and b be two sentences in the sense of mathematical :a _ b logic. It is known that the composite statement is false. For each a and b determine whether it is true or false. :a =):b The statement a is true, b is false. is false. For each a and b determine whether it is true or false. The statement a is false, b is true. Both statements are true. The statement a is false, b is true. Both statements are true. The statement a is true, b is false.
    [Show full text]
  • Whole Numbers
    Unit 1: Whole Numbers 1.1.1 Place Value and Names for Whole Numbers Learning Objective(s) 1 Find the place value of a digit in a whole number. 2 Write a whole number in words and in standard form. 3 Write a whole number in expanded form. Introduction Mathematics involves solving problems that involve numbers. We will work with whole numbers, which are any of the numbers 0, 1, 2, 3, and so on. We first need to have a thorough understanding of the number system we use. Suppose the scientists preparing a lunar command module know it has to travel 382,564 kilometers to get to the moon. How well would they do if they didn’t understand this number? Would it make more of a difference if the 8 was off by 1 or if the 4 was off by 1? In this section, you will take a look at digits and place value. You will also learn how to write whole numbers in words, standard form, and expanded form based on the place values of their digits. The Number System Objective 1 A digit is one of the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. All numbers are made up of one or more digits. Numbers such as 2 have one digit, whereas numbers such as 89 have two digits. To understand what a number really means, you need to understand what the digits represent in a given number. The position of each digit in a number tells its value, or place value.
    [Show full text]
  • Long Division, the GCD Algorithm, and More
    Long Division, the GCD algorithm, and More By ‘long division’, I mean the grade-school business of starting with two pos- itive whole numbers, call them numerator and denominator, that extracts two other whole numbers (integers), call them quotient and remainder, so that the quotient times the denominator, plus the remainder, is equal to the numerator, and so that the remainder is between zero and one less than the denominator, inclusive. In symbols (and you see from the preceding sentence that it really doesn’t help to put things into ‘plain’ English!), b = qa + r; 0 r < a. ≤ With Arabic numerals and using base ten arithmetic, or with binary arith- metic and using base two arithmetic (either way we would call our base ‘10’!), it is an easy enough task for a computer, or anybody who knows the drill and wants the answer, to grind out q and r provided a and b are not too big. Thus if a = 17 and b = 41 then q = 2 and r = 7, because 17 2 + 7 = 41. If a computer, and appropriate software, is available,∗ this is a trivial task even if a and b have thousands of digits. But where do we go from this? We don’t really need to know how many anthills of 20192 ants each can be assembled out of 400 trillion ants, and how many ants would be left over. There are two answers to this. One is abstract and theoretical, the other is pure Slytherin. The abstract and theoretical answer is that knowledge for its own sake is a pure good thing, and the more the better.
    [Show full text]
  • Long Division
    Long Division Short description: Learn how to use long division to divide large numbers in this Math Shorts video. Long description: In this video, learn how to use long division to divide large numbers. In the accompanying classroom activity, students learn how to use the standard long division algorithm. They explore alternative ways of thinking about long division by using place value to decompose three-digit numbers into hundreds, tens, and ones, creating three smaller (and less complex) division problems. Students then use these different solution methods in a short partner game. Activity Text Learning Outcomes Students will be able to ● use multiple methods to solve a long division problem Common Core State Standards: 6.NS.B.2 Vocabulary: Division, algorithm, quotients, decompose, place value, dividend, divisor Materials: Number cubes Procedure 1. Introduction (10 minutes, whole group) Begin class by posing the division problem 245 ÷ 5. But instead of showing students the standard long division algorithm, rewrite 245 as 200 + 40 + 5 and have students try to solve each of the simpler problems 200 ÷ 5, 40 ÷ 5, and 5 ÷ 5. Record the different quotients as students solve the problems. Show them that the quotients 40, 8, and 1 add up to 49—and that 49 is also the quotient of 245 ÷ 5. Learning to decompose numbers by using place value concepts is an important piece of thinking flexibly about the four basic operations. Repeat this type of logic with the problem that students will see in the video: 283 ÷ 3. Use the concept of place value to rewrite 283 as 200 + 80 + 3 and have students try to solve the simpler problems 200 ÷ 3, 80 ÷ 3, and 3 ÷ 3.
    [Show full text]
  • "In-Situ": a Case Study of the Long Division Algorithm
    Australian Journal of Teacher Education Volume 43 | Issue 3 Article 1 2018 Teacher Knowledge and Learning In-situ: A Case Study of the Long Division Algorithm Shikha Takker Homi Bhabha Centre for Science Education, TIFR, India, [email protected] K. Subramaniam Homi Bhabha Centre for Science Education, TIFR, India, [email protected] Recommended Citation Takker, S., & Subramaniam, K. (2018). Teacher Knowledge and Learning In-situ: A Case Study of the Long Division Algorithm. Australian Journal of Teacher Education, 43(3). Retrieved from http://ro.ecu.edu.au/ajte/vol43/iss3/1 This Journal Article is posted at Research Online. http://ro.ecu.edu.au/ajte/vol43/iss3/1 Australian Journal of Teacher Education Teacher Knowledge and Learning In-situ: A Case Study of the Long Division Algorithm Shikha Takker K. Subramaniam Homi Bhabha Centre for Science Education, TIFR, Mumbai, India Abstract: The aim of the study reported in this paper was to explore and enhance experienced school mathematics teachers’ knowledge of students’ thinking, as it is manifested in practice. Data were collected from records of classroom observations, interviews with participating teachers, and weekly teacher-researcher meetings organized in the school. In this paper, we discuss the mathematical challenges faced by a primary school teacher as she attempts to unpack the structure of the division algorithm, while teaching in a Grade 4 classroom. Through this case study, we exemplify how a focus on mathematical knowledge for teaching ‘in situ’ helped in triggering a change in the teacher’s well-formed knowledge and beliefs about the teaching and learning of the division algorithm, and related students’ capabilities.
    [Show full text]
  • The Euclidean Algorithm
    1. The Euclidean algorithm. The euclidean algorithm is used to find GCD(a; b), the greatest common divisor of two positive integers a; b 2 N, and can be used to find m; n 2 Z so that: GCD(a; b) = ma + nb: In particular, if p is a prime, GCD(a; p) = 1, so writing 1 = ma + np we −1 find that in the field Zp (of residues mod(p)) we have: (¯a) =m ¯ . ¯ Example. Find the multiplicative inverse of 11 in Z37. The euclidean algorithm gives: 37 = 3 × 11 + 4; 11 = 2 × 4 + 3; 4 = 3 + 1: Thus: 1 = 4 − 3 = 37 − 3 × 11 − [11 − 2 × (37 − 3 × 11)] = 3 × 37 − 10 × 11; ¯ −1 ¯ ¯ or m = −10; n = 3. Thus (11) = −10 = 27 in Z37. ¯ −1 Problem 1. Find (4) in the field Z31. 2. Long division. Suppose x 2 [0; 1) has an eventually periodic decimal representation: x = 0:m1m2 : : : mN d1d2 : : : dT : Then: N+T N 10 x − m1 : : : mN d1 : : : dT = 10 x − m1 : : : mN = 0:d1 : : : dT : Hence: N T 10 (10 − 1)x = m1 : : : mN d1 : : : dT − m1 : : : mN := m 2 N; or: m d : : : d x = (x = 1 T if the expansion is periodic): 10N (10T − 1) 10T − 1 In other words: any real number (in [0; 1)) with an eventually periodic decimal expansion is rational, and can be written in the form m=q, where q is a difference of powers of 10. (Or, considering base 2 expansions, with q a difference of powers of two.) 1 And yet any rational number has an eventually periodic decimal expan- sion! This can be established by considering long division.
    [Show full text]
  • Solve 87 ÷ 5 by Using an Area Model. Use Long Division and the Distributive Property to Record Your Work. Lessons 14 Through 21
    GRADE 4 | MODULE 3 | TOPIC E | LESSONS 14–21 KEY CONCEPT OVERVIEW Lessons 14 through 21 focus on division. Students develop an understanding of remainders. They use different methods to solve division problems. You can expect to see homework that asks your child to do the following: ▪ Use the RDW process to solve word problems involving remainders. ▪ Show division by using place value disks, arrays, area models, and long division. ▪ Check division answers by using multiplication and addition. SAMPLE PROBLEM (From Lesson 21) Solve 87 ÷ 5 by using an area model. Use long division and the distributive property to record your work. ()40 ÷+54()05÷+()55÷ =+88+1 =17 ()51×+72= 87 Additional sample problems with detailed answer steps are found in the Eureka Math Homework Helpers books. Learn more at GreatMinds.org. For more resources, visit » Eureka.support GRADE 4 | MODULE 3 | TOPIC E | LESSONS 14–21 HOW YOU CAN HELP AT HOME ▪ Provide your child with many opportunities to interpret remainders. For example, give scenarios such as the following: Arielle wants to buy juice boxes for her classmates. The juice boxes come in packages of 6. If there are 19 students in Arielle’s class, how many packages of juice boxes will she need to buy? (4) Will there be any juice boxes left? (Yes) How many? (5) ▪ Play a game of Remainder or No Remainder with your child. 1. Say a division expression like 11 ÷ 5. 2. Prompt your child to respond with “Remainder!” or “No remainder!” 3. Continue with a sequence such as 9 ÷ 3 (No remainder!), 10 ÷ 3 (Remainder!), 25 ÷ 3 (Remainder!), 24 ÷ 3 (No remainder!), and 37 ÷ 5 (Remainder!).
    [Show full text]
  • Blind Mathematicians
    The World of Blind Mathematicians A visitor to the Paris apartment of the blind geome- able to resort to paper and pencil, Lawrence W. ter Bernard Morin finds much to see. On the wall Baggett, a blind mathematician at the University of in the hallway is a poster showing a computer- Colorado, remarked modestly, “Well, it’s hard to do generated picture, created by Morin’s student for anybody.” On the other hand, there seem to be François Apéry, of Boy’s surface, an immersion of differences in how blind mathematicians perceive the projective plane in three dimensions. The sur- their subject. Morin recalled that, when a sighted face plays a role in Morin’s most famous work, his colleague proofread Morin’s thesis, the colleague visualization of how to turn a sphere inside out. had to do a long calculation involving determi- Although he cannot see the poster, Morin is happy nants to check on a sign. The colleague asked Morin to point out details in the picture that the visitor how he had computed the sign. Morin said he must not miss. Back in the living room, Morin grabs replied: “I don’t know—by feeling the weight of the a chair, stands on it, and feels for a box on top of thing, by pondering it.” a set of shelves. He takes hold of the box and climbs off the chair safely—much to the relief of Blind Mathematicians in History the visitor. Inside the box are clay models that The history of mathematics includes a number of Morin made in the 1960s and 1970s to depict blind mathematicians.
    [Show full text]