Which of the following generalizations seem to be true when we compare and contrast paper 1 mathematicians (women and minorities in the 18th and 19th centuries) to paper 2 mathematicians (women and minorities born between 1900 and 1925.

1. In paper 1, all but one of the women we studied were unmarried (and the married woman had gotten married in order to travel and obtain an education). In paper 2, most of the women we studied were married, and some had children. It seems as time is progressing, women are better able to balance a career with family.
2. Fifty percent of the women that we studied in paper 2 were married to mathematicians, and so they would have been subject to nepotism rules. During the day that we reviewed for this WebCt quiz, it was also true that fifty percent of the women mathematicians present were married to mathematicians. While we do not expect this to be the percentage in general, it makes sense that many woman mathematicians have married male mathematicians, since their social circle would often consist of many mathematicians and scientists. Studies have backed up this phenomenon.
3. In paper 1, the two African Americans with PhDs ended up with positions at Howard University, and it is still true in paper 2 that most of the African American mathematicians ended up with positions at traditionally Black institutions.
4. We have seen a number of women mathematicians that had positions at women's colleges.
5. A number of African American PhDs received their doctorates from the University of Michigan.
6. Two African American mathematicians we have seen worked with difference equations, while two famous women mathematicians we have see worked with partial differential equations. It seems that there might be certain areas in which women and minorities specialize.
7. In paper 1, we studied early women who were teachers, and as time progressed, we saw that women mathematicians became leaders in their fields. In paper 2, we see some women and African American mathematicians that are leaders in their fields, and others who have done significant mathematics to obtain a PhD, but may focus on other things, such as teaching, afterwards.
8. In paper 1, African American mathematicians and women seemed to be oddities in the sense that there were so few of them. The numbers of African American and women mathematicians have increased greatly as time has progressed. Part of this increase is probably due to increased educational opportunities.
9. We have seen numerous examples of blatent gender and racial discrimination in both papers 1 and 2.
10. Most of the mathematicians that we studied had a good support system from family and/or mentors, which seems to have been important to help them overcome other barriers.

Which of the following are true about David Blackwell and his mathematics?

1. He is considered one of the greatest African American mathematicians that has ever lived.
2. Blackwell was lucky to attend a mixed school instead of the all African American schools that were the norm in most areas of the nation at that time. Blackwell states, 3Southern Illinois was probably fairly racist. But I was not even aware of these problems -- I had no sense of being discriminated against.2
3. After graduation, he was appointed a Postdoctoral Fellow at the Institute for Advanced Study (IAS) for a year. While every IAS fellow was automatically appointed at Princeton University also, the president of Princeton was upset and admonished IAS for allowing an African American on the faculty.
4. Blackwell is known for his work on statistics and game theory, which have many real-life applications.
5. If a jar contains seven ping-pong balls, numbered 1 through 7, and two balls are drawn from the jar (the first ball is not replaced before the second is drawn), then the probability that both balls have even numbers on them is 2/7.
6. If a jar contains seven ping-pong balls, numbered 1 through 7, and two balls are drawn from the jar (the first ball is not replaced before the second is drawn), then the probability that one ball has a 7 on it is 2/7.
7. In Blackwell's paper on The Big Match, every day player 2 chooses a number,0 or 1, and player 1 tries to predict player 29s choice, winning a point if he is correct. This continues as long as player 1 predicts a 0. But if he ever predicts a 1, all future choices for both players are required to be the same as that day9s choices; if he is wrong on that day, he wins zero every day thereafter.
8. The game is called the Big Match since whoever has won when player 1 guesses a 1 wins "Big", since they win from that point everafter.
9. Even if player 1 has been winning a lot with the guess of 0, it doesn't make sense to guess a 1, since there is a very big risk with guessing wrong.
10. Even if player 1 has been winning a lot with the guess of 0, there is a 50% chance that player 2 will next choose a 1 if player 2 is using a coin.
11. If player 1 has been winning a lot with the guess of 0, there is a much greater chance that player 2 will next choose a 1 if player 2 is using a coin.
12. If player 2 is using a coin, there is a small chance that player 1 will win many times in a row.

Which of the following are true about Marjorie Lee Browne and her mathematics?

1. Her father Lawrence Lee, a railway postal clerk, had a great love for mathematics. He was known around Memphis as a whiz at mental math. Marjorie has been quoted to say that she knew she loved math even from an early age, mostly because of the enthusiasm her father shared with her.
2. She obtained her Ph.D. from the University of Michigan in 1949, becoming one of the first two African-American woman to receive a doctorate degree in mathematics.
3. She was unable to find a job at a research university. It is possible that this rejection was a result of racial discrimination in the United States at that time.
4. She began teaching at North Carolina Central University in Durham, North Carolina, where she remained until she retired in 1979. She was the only PhD faculty member in the mathematics department. Because of this, one could say that she was underemployed. On the other hand, she loved teaching and was extremely interested and involved in the education of secondary education teachers.
5. Her only published mathematical article is 3A Note on the Classical Groups.2
6. Marjorie was motivated by her love of math, her desire to instill in her students the same passion that she had for math, and her desire to see her students achieve not only in the math classroom, but also in life (Morrow, 21). She had a genuine love not only for math but for her students as well. Her philosophy was that 3You appreciate those things in life that you earn. People can not take those things away from you.2 (24). Shortly before her death she was quoted as saying, 3If I had my life to live again, I wouldn9t do anything else. I love mathematics.2
7. GL(n,C) is the set of nxn complex matrices that are invertible.
[i 1]
[1 i]
is an element of GL(2,C).
[i 1]
[-1 i]
is an element of GL(2,C).
9. SL(n,C) consists of all nxn matrices with determinant equal to one. These matrices preserve volume, orientation, and distance between points in R^n.
10. GL(n,C) can be decomposed as the topological product of C with SL(n,C) via the following:
Take any matrix in GL(n,C) as [a_ij] where i and j run from 1 to n.
We know the determinant of this matrix is non-zero, since it is invertible, so let d be this determinant. Notice that d can be any non-zero complex number.
We identify the complex number d with the matrix that is the nxn identity matrix everywhere except in the a_11 slot, where we replace 1 by d. The map from C to these matrices is continuous, as is the inverse map, and so topologically, these are equivalent.
To come up with a matrix in SL(n,C) that completes the topological decomposition, replace the first row of our original GL(n,C) matrix by dividing each entry in the first row by d. Leave the rest of the matrix the same. We can do this since d is not 0. In addition, the determinant of this matrix is now 1, and so it is an element of SL(n,C).
This gives us the topological decomposition.

Which of the following are true based on Grace Murray Hopper and her mathematics?

1. Her mother gave her a love for mathematics, as well as her grandfather who gave Grace her first lessons in angles and curves.
2. She was known for her work in computer science, but did graduate with a degree in mathematics.
3. During her work on the Mark I, Hopper was given credit for coining the term 3bug2, which is a reference to a glitch in the computer. She actually found a moth inside the computer, which was causing the problems.
4. With the outbreak of World War II, Hopper made a life-altering decision to serve her country by joining the Navy. The process was not an easy one. At age 34, weighing 105 pounds, she was considered overage and underweight for military enlistment. In addition, her position as a mathematics professor was declared crucial to the war effort. Navy officials asked her to remain a civilian. These obstacles did not stop Grace Hopper. She obtained a waiver for the weight requirement, special government permission, and a leave of absence from Vassar College. In December 1943, she was sworn into the U.S. Naval Reserve. She went on to train at Midshipman's School for Women, graduating first in her class.
5. She earned a Ph.D. from Yale in 1934 (with a thesis on "New Types of Irraducibility Criteria"). In 1936 she published a paper on "The ungenerated seven as an index to Pythagorean number theory" in the American Mathematical Monthly.
6. x^2+ x + (1/2) is an irreducible polynomial over the rational numbers.
7. x^2-(1/2) is a reducible polynomial over the rational numbers.
8. When the degree of a polynomial gets large, it is hard to tell whether it will factor or not. Eisenstein's criterion will work on any polynomial to tell us whether it is reducible over the rational numbers or not.
9. x^20 +5*x^19 + 25*x^18+ 20*x ^17 +5 is irreducible by Eisenstein's criterion.
10. Sometimes, even if Eisenstein's criterion doesn't directly apply, one can use a substitution trick to show that it is irreducible. For example, if F(x)=x^4 + 1, then we cannot apply Eisenstein's criterion to F(x) directly, but we can apply it to g(x) = f(x+1), since this reduces to x^4 + 4 x^3 + 6 x^2 + 4x + 2. Since g(x) is irreducible by Eisenstein's with p=2, we see that f(x+1), and hence f(x) is also irreducible.
11. In Hopper's thesis, she worked to determine the reducibility or irreducibility of polynomials via first converting them to geometric figures and then determining whether the geometric figures decompose. For example, we could decompose an icosahedron into 10 tetrahedrons. So, if our polynomial converted to an icosahedron, then it would be reducible, since the icosahedron decomposes into smaller figures.

Which of the following are true based on Olga Ladyzhenskaya and her mathematics

1. While we do not know much about her life, we do know that she received encouragement from her father and many of her colleagues. Her work on Navier Stokes equations has had a profound effect on fluid dynamic
2. Sofia Kovalevsky, from paper 1, was another woman mathematician who worked in partial differential equations in Russia, although she lived in the 19th century. She was determined to continue her education at the university level. However, the closest universities open to women were in Switzerland, and young, unmarried women were not permitted to travel alone. To resolve the problem Sofia entered into a marriage of convenience to her husband
3. The last Tsar of Russia abdicated the throne in February of 1917. With the fall of the old regime, many old gender barriers fell, as well. The period after the Bolsheviks rose to power was a time of many changes for all Russians, but none were more affected than the women of the time. Lenin, the leader of the Bolshevik party (later called the Communists) was greatly disturbed by the domestic enslavement of Soviet women, and almost immediately granted political equality for females throughout the nation. With this newfound freedom, women were presented with many new opportunities in all aspects of life, and many challenges, as well. Lenin reformed many civil and penal codes to the advantage of women. Almost overnight all learning institutions opened their doors to both sexes, which suddenly gave women the opportunity to strive for professional careers and higher paying jobs. Women were given equal standing in marriage, and it became possible for them to get divorced, to have abortions, and to sue for child support. Women could own property. Within the Communist party, women rose to leadership positions. In theory, there was complete equality.
4. A PDE is an equation that is written in terms of partials of functions. f(x,y)=sin(xy) is a solution to the PDE x^2 *f +f_(yy)=0
5. u(x,t) = e^(-lambda^2 * t) * B sin (x) is a solution to the PDE u_t = u_xx
6. u(x,t) = e^(-lambda^2 * t) *[ A cos(lambda x) + B sin (lambda x)] is a solution to the PDE u_t = u_xx. When we apply the boundary value u(0,t) to this solution, we obtain A=0.

Which of the following are true based on Julia Robinson and her mathematics?

1. Despite that fact that she received the highest grades in high school classes, Julia was never encouraged by any of her teachers to pursue advanced mathematics.
2. While Julia was taking math classes at Berkley, she met Ralph Robinson who not only was one of her professors, but also eventually became her husband. Julia mentioned that she would not have become a mathematician if it were not for her husband Ralph and his great encouragement.
3. Because of a rule preventing members of the same family from teaching in the same department, Julia Robinson could not continue to work in her teaching assistantship in the mathematics department
4. She said "What I really am is a mathematician. Rather than being remembered as the first woman this or that, I would prefer to be remembered as a mathematician should, simply for the theorems I have proved and the problems I have solved."
5. Julia contributed significantly to the solution of Hilbert's 10th problem: Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in integers.
6. Fermat's Last Theorem can be written as a Diophantine equation via x^n +y^n -z^n =0 for n>2.
7. The equation x^2 + y^2 -z^2 = 0 has infinitely many integer solutions.
8. Sometimes, as in the case of Fermat's Last Theorem, people work for hundred's of years and use many different techniques in order to determine the solvability of an equation.
9. In Fermat's Last Theorem, there are no integer solutions to the equation.
10. The solvability of diophantine equations has applications to coding. If you can come up with an equation that is hard to solve in a finite number of steps, you can use that equation to code in different commands and information that you do not want the general public to know. In order to decode the message you would have to have prior knowledge of what was put into the original equation.
11. While for many diophantine equations, we can eventually determine whether the equation is solvable in integers or not, we will never be able to find a method that will work to determine integer solvability for all Diophantine equations.

Which of the following are true based on Alice Schafer and her mathematics?

1. After her parents died, her aunts took care of her and supported her decision to do mathematics. She later married a mathematician.
2. She faced numerous instances of gender discrimination. When asked to write a letter of recommendation for Alice, her high school principle replied 3girls shouldn9t do math2, and never sent the letter. Women were not allowed in the library at the University of Richmand. As a result, every time Alice wanted to do research she had to order the book she needed and had to read it in a study room designated just for women. She also had a college professor who was known to say that he wanted to fail every woman that attended his classes.
3. Schafer faced her discrimination head on by fighting against it. At the University of Richmond she was mainly responsible for opening the library to women. Sadly, she was kicked out on her first day in the library for laughing out loud while reading a book. Alice also won the Crump prize which her discriminatory college professor took part in grading. She also helped the start of the AWM (Association for Women in Mathematics) and then was pronounced the second president of the organization in 1972. She took part as an active member from the start and is even a member now.
4. The oscillating plane is one idea that arose in one of her papers. It is the plane formed by the tangent (first derivative) and normal vectors (second derivative) to a curve. It provides us with geometric intuition about how a curve changes, which can be useful in real life.
5. Given F(t)=(a cos(t), a sin(t), bt), where t is an element of the reals, the graph of F(t) is a spiral on the surface of a cylinder of radius a^2. If the spiral is a slinky, then b tells us how much we are stretching the slinky in the z direction.
6. Given F(t)=(a cos(t), a sin(t), bt), the oscillating plane is formed by the vectors (-a sin(t),a cos(t),b) and (-a cos(t), a sin(t), 0).
7. Given F(t)=(a cos(t), a sin(t), bt), if a is a very small number, then the oscillating planes on that curve will turn much faster (with t) than with a curve which a much larger a value.
8. The oscillating plane is not defined at points where the first or second derivatives are not defined, for example, as on a curve going through one of the sharp ends of a football (which does not backtrack). We can see this as the oscillating planes approaching a sharp end of the football from one side do not converge to the same plane as the oscillating plane approaching from the other side.

Which of the following are true based on Clarence Stephens and his mathematics?

1. The support of his family and encouragement from two of his teachers was instrumental in his obtaining a PhD in mathematics. While Stephens was attending school at JCSU, his only plans were to become a high school math teacher. But, the Dean of the College of Liberal Studies, T. E. McKinney, had returned from a visit to the University of Michigan, saw Stephens one day during his senior year at school, and said to him that the University of Michigan was where he needed to go. So Stephens went to the University of Michigan for his master9s degree. While he planned to be a lawyer, while talking to one of his professors at Michigan, George Rainich, Stephens was encouraged to return to school in the doctoral program. Having a professor assume his return to further his education opened Stephens9 eyes to his potential. Professor Rainich9s comments were so inspiring that Stephens set more personal goals that he could achieve in the mathematical field.
2. During the time when Stephens was in school, he had to wait tables to pay for tuition, as there were no teaching assistant positions for African Americans. He also worked as a deliveryman for a local drug store that paid six dollars a week.
3. Stephens took a position as a professor of mathematics at Prairie View A. and M. College in Texas in 1946, and mentored Beauregard Stubblefield while he was there.
4. Stephens is best known for his teaching method which is known for producing mathematics majors many of whom go on to graduate school. He has said: "More than fifty years ago I came to the conclusion that every college student who desired to learn mathematics could do so. I spent my entire professional life believing that this was the case." His teaching methods can be summarized as methods for developing and supporting these abilities in students.
5. Difference equations are equations in terms of the differences of functions. Difference equations are very useful in real life since they can be used in situations of discrete data that prevents us from taking the derivative. Instead, we can take the divided difference representing the slope of the secant line, which gives an approximation to the derivative.
6. A non-linear difference equation is a difference equation where the differences of the functions are given as constants times the functions.
7. A linear difference equation that could be used for a traffic flow problem is N_(t)=N_(t-1) +F - G, where F is the number of cars coming in, and G is the number leaving.
8. Nonlinear difference equations are very complicated equations which, however, have many practical uses. Difference equations can be used to describe things such as changes of the texture of oil under the differing conditions of heat and pressure in an engine. They also can be used to model certain interactions of individuals such as the spread of rumors and sicknesses, and natural growth such as the growth colonies in species populations (Bauldry). Difference equations are used to model discrete data such as traffic patterns and temperature measurements.
9. If P is the total population, and the number of people who know the rumor after n-1 time intervals is a_n, then the equation a_n = a_(n-1) + c * a_(n-1) * (P - a_(n-1)) is a non-linear difference equation that can be used to model the spread of a rumor. The c * a_(n-1) * (P - a_(n-1)) term makes sense since the rumor will be spread by some percentage, c, of the interaction between the number of people that know the rumor and those that don't.

Which of the following are true based on Beauregard Stubblefield and his mathematics?

1. As a child, Stubblefield9s father instilled in him the desire to pursue mathematics. Every week, he would have Beauregard calculate the grocery bill, claiming each time that his calculations were correct. His father would then tell him that because of this, he would turn out to be a good mathematician.
2. An appropriate quote that describes Stubblefield is that "Success doesn9t come to you you go to it." One example of this is when he did not receive a reply from three of the four graduate schools that he applied to. Because Stubblefield was determined to go to the University of Michigan, he decided to drive to the campus to find out why he did not receive an answer. Surprised as to the boldness of this man, the University of Michigan accepted him into their graduate program.
3. Dr. Richardson helped in bringing the first African American woman to the masters in mathematics program here at Appalachian State University in 1966. He talks about how when she went to look for housing, she was told that each apartment was already promised or filled. 3Stubblefield probably would have encountered the same thing, 3 Richardson says. The university, knowing this, decided to put Beauregard in faculty housing when he came here in 1971.
4. For his paper entitled "Lower bounds for odd perfect numbers (beyond the googol)", written while he was here at Appalachian State University, Stubblefield used up a lot of computer time.
5. A perfect number is a number that is divisible by only 1 and itself. 6 is a perfect number but prime numbers are never perfect.
6. Every even perfect number is of the form 2^(k-1)*(2^k-1) for some k>1 so that (2^k-1) is prime.
7. We know that there are infinitely many even perfect numbers since there are infinitely many k's that we can plug in to the formula.
8. We do not know if ANY odd perfect numbers exist. If they do exist, then the work of Stubblefield and other mathematicians show us that they must be very large numbers.
9. The existence of perfect numbers have applications to string theory in physics.

Which of the following are true based on Olga Taussky-Todd and her mathematics?:


1. Her father was an industrial chemist and a journalist. He wanted his children to receive a good education and go into the field of arts. Instead, all three of his daughters went into the fields of science and mathematics. Her mother was also very encouraging.
2. By the age of fifteen she was enrolled in the gymnasium, the only school for girls to choose from. They were taught Latin instead of science or mathematics, and so she educated herself in mathematics and science.
3. At one interview, she was asked: "I see you have written several joint papers. Were you the senior or the junior author?" Another member of the committee was G.H. Hardy, who interjected, "That is a most improper question. Do not answer it!" At another interview she was asked, "I see you have collaborated with some men, but with no women. Why?" Olga replied that that was why she was applying for a position in a women's college! It is less amusing to learn that the senior woman mathematician insisted that woman students not do their theses with Olga, even when male colleagues considered her the most suited to the projected research, because it would be damaging to their career to have a woman supervisor.
4. Olga and her husband, also a mathematician, were invited by Caltech to join the faculty in 1957. The offer was (as was usual at the time) for the husband to become Professor and the wife Research Associate; but their offices were adjacent and the same size, and Olga was welcome to conduct seminars and supervise theses. As she did. The anomaly in their status, due to a nepotism law preventing relatives from working in the same department, ceased to look ideal when, in 1969, a very young Assistant Professor of English was glorified by the press as the first woman ever on Caltech's faculty. The first, indeed! What about Olga? This did rub her the wrong way; she went straight to the administration and had her rank changed. Effective in 1971, she was Professor.
5. Flutter is a phenomena in flight that is related to the interaction of the elastic forces in the airframe of an aircraft and the aerodynamic forces being exerted on the aircraft. The combined effect causes an induced self-excited vibration that is unstable above a certain speed. The flutter speed of an aircraft must be calculated before the aircraft is built and flown. At the time, the calculations were being done on hand operated machines by young girls. She found a way to simplify the process by first looking for the eigenvalues using the Gersgorin theorem She was working on a 6 x6 matrix. The flutter parameter, . She used matrix entries around twenty, but could not find the eigenvalues. Then she used the Gersgorin theorem to narrow down the possible range of answers, reducing the amount of calculations considerably.
6. Within a given complex matrix, the Gershgorin theorem can be used to find the location of the eigenvalues, which will be enclosed within the Gersgorin circle. Given any row in a matrix, the entry a_ii, is the center of the circle. The rest of the entries in that row are added up to get the radius of the circle. This circle will contain some of the eigenvalues for the given matrix. The union of all the circles will contain all of the eigenvalues for any given matrix.
7. To find the eigenvalues of a matrix, we must solve the equation det( lambda I - A) =0 for I.
8. Given the matrix
[ i 1]
[ 1 i]
if we solve for the eigenvalues by hand, we obtain complex eigenvalues i-1 and i+1.
If we apply the Gershgorin circle theorem to this matrix, we obtain the result that each eigenvalue will be contained in a circle with center (0,i) in the complex plane, and radius 1.
In this example, the eigenvalues are right on the boundary of the circle.
9. Given the matrix
If we solve for the eigenvalues by hand, then we obtain eigenvalues of approximately 1.791287848 + i and -2.791287848 +i.
The picture at the top of the page accurately shows the Gershgorin circles. The centers of the circles are accurately marked by diamonds
The actual eigenvalues are shown to be within the circles and are marked by squares. The Gershgorin theorem doesn't give use the actual eigenvalues, just an estimate of where they are located.

Which of the following are true based on J. Ernest Wilkins, Jr. and his mathematics?

1. The 1920s were a time in the history of the United States of America often looked upon as a decade of luxury and success, but this was not the case for all it9s citizens. In fact, in 1923, when he was born, 29 African-Americans were lynched in this country. Throughout the United States, African-Americans were forced to experience grave injustices because of the color of their skin. Very few African-Americans were able to rise above this discrimination and succeed in the academic and especially the mathematical world. One such man was J. Ernest Wilkins.
2. It can be inferred from the small amount of information available about the parents of Wilkins, Jr., that most likely the importance of learning and education was stressed in the home. During his teenage years, Wilkins, Jr. was credited by national newspapers as 3the negro genius2
3. It was difficult for Wilkins to find a job at a research university. It is possible that this rejection was a result of racial discrimination in the United States at that time.
4. Another experience that allows us to understand more about the obstacles and discriminations Wilkins faced as an African-American mathematician, was described by Lee Lorch, a Caucasian-American human rights activist, in 1947: "Wilkins was a few years past the Ph.D. ... He received a letter from the AMS Associate Secretary for that region urging him to come and saying that very satisfactory arrangements had been made with which they were sure he'd be pleased: they had found a ``nice colored family" with whom he could stay and where he would take his meals! The hospitality of the University of Georgia (and of the American Mathematics Society) was not for him - he refused. This is why the meeting there was totally white." Because of this encounter, Wilkins has never since attended a meeting of the American Mathematical Society in the Southeast.
5. Wilkins published many papers and is best known for his work on the development of shielding against gamma radiation. Two of his papers were on ruled and cubic surfaces.
6. One important fact about ruled surfaces is that they can be generated by straight lines. One would never know this from looking at the surface or its usual equation in terms of x, y and z coordinates, but ruled surfaces can all be rewritten to highlights the generating lines. A practical application of ruled surfaces is that they are used in civil engineering. Since building materials such as wood are straight, they can be thought of as straight lines. The result is that if engineers are planning to construct something with curvature, they can use a ruled surface since all the lines are straight.
7. x(t, k)=(cos(t), sin(t), 0) + k* (-sin(t), cos(t), 1) gives a ruling for a hyperboloid.
8. x(t,k ) = (cos(t), sin(t), 0) +k* (0,0,1) gives a ruling for a cylinder.
9. x(t,k ) = (0,0,0) + k* (cos(t), sin(t), 1) gives a ruling for a cone.
10. The hyperboloid, cylinder and cone are the only ruled surfaces.
11. A cubic surface is a surface defined by a degree 3 polynomial in three dimensional space. Surfaces become more complicated as you increase the degree of the defining polynomial. For degree 1, the surface is just a plane. Degree 2 surfaces are more interesting but also fairly easy to understand. To understand degree 3 surfaces, geometers had to develop new techniques which have since been used to study surfaces of higher degree as well as higher dimensional shapes. One can use the algebra of polynomials (algebraic geometry) to study their geometric properties. One interesting fact is that every cubic surface has exactly 27 straight lines on it! However, the points on these lines may not always have real coordinates-- to "see" the lines, you may have to consider the polynomial equation in 3 dimensional complex space!