Zvezdelina Entcheva Stankova

Visiting Professor


713 Evans Hall

Department of Mathematics

University of California at Berkeley, CA 94720-3840

Tel: 510-642-3768, Email: "last name, lowercase" at "university name" dot edu

Office Hours: TTh 9:30-10:30am (first in front of 155 Dwinnelle right after lecture, then Evans 713), T 3:30-4:30pm in Evans 713.

Webpage: http://math.berkeley.edu/~stankova

Course Syllabus for MATH 53, Spring 2016

(updated 1/22/2015)

GSIs finalized office hours will be revised here. Check them out and keep them with you at all times. You are welcome to visit any GSI's office hours, along with the professor's office hours.

Christopher Eur's revised office hours (as of 1/24/2016): M 5-6pm, T 12:30-1:30pm, W 11am-12noon.

Kai-Chieh Chen's revised office hours (as of 1/25/2016): M 2:30-4pm, W 3-4:30pm.

Andrew Dudzik's revised office hours (as of 3/14/2016): M 1-2pm, W 1-2pm, F 11am-12noon.

Christopher Eur: additional office hours on Monday 4/5/2016 from 3:30-6pm.

All other GSI's office hours are as in the syllabus.

For enrollment questions:

Do NOT contact the instructor or the GSIs. We have no control over enrollment. Contact instead Thomas Brown, 965 Evans, 10:30am-12noon or 1-5 pm. You need to see him in person (not by email!) to resolve enrollment questions. Students who wish to switch sections with open spaces can use the "Switch Section" link in TeleBears. Thomas Brown cannot move students into sections that are full. However, he can often swap students between sections. If two students are willing to swap sections, both students need to go to Thomas and he will try to switch them.

Email is ONLY for emergencies (e. g., medical and family emergencies).

Email is NOT for resolving enrollment questions, or asking for letters, or for discussion of midterm results, or for any discussion about how the student is doing in the class or how to improve. For such questions, students need to come see me in person in office hours (bringing all necessary documentation with you). You are also welcome to visit any GSI's office hours to discuss math questions or how to improve in the course; the GSIs are very qualified to discuss any math question. This is clearly written in the syllabus and is discussed in detail during the first lecture.

Homework Assignments, MATH 53

HW1A. Read: 10.1. Curves defined by parametric equations. 10.5. Conic sections (concentrate on recognizing the type of conic section: parabola, ellipse, or hyperbola, and sketching graphs)

Write Exercises:

10.1. #6,8,10,12,14,16,20,22,24,38. Bonus: #31,46(a)(c).

10.5. Just sketch the graphs: #6,14,22. Bonus: #8,16,24 (complete the square(s) in the bonus problems).

HW1B: Read: 10.2. Calculus with Parametric Curves: Tangents (in polar coordinates), Examples 1 and 2, pp. 650-651. 10.3. Polar Coordinates

Write Exercises:

10.2. #2,6,8,14,18 Bonus: #28,30

10.3. #2,4,6,10,16,18,24,30,56,60 Bonus: #14,28

Notes from class 1/21/16 and office hours:

1. The "mystery" of why the "polar" angle theta for point A(7 cos(45), 5 sin(45)) on our ellipse did not match the "parameter" angle t? There is no mistake in what we did in class. Indeed, the polar angle theta for A is arctan(y/x)=arctan(5/7). This angle theta is different from t = 45 degrees = pi/4 for the simple reason that the parameter t is different from the angle theta. When we parametrized the ellipse x^2/49 + y^2/25 = 1, we used the original parameter t from the unit circle. However, when the unit circle is rescaled to the ellipse, i.e., (u,v) --> (x=7u, y=5v), the angle with the x-axis changes. In our example, take point B(cos(45),sin(45)) on the unit circle: the angle (whether "polar" or "parameter") is 45 degrees; yet, when we rescale to get from point B to point A, the angle with the x-axis changes to theta = arctan(5/7), which is the polar angle and not the original 45-degree angle used as a parameter.

2. Regarding the polar equation of the line y = (square root of 3) x. If we want one polar equation, then we have to allow r to be negative in r = pi/3. If we want to keep r non-negative, we will need two polar equations: r = pi/3 and r = 4 pi/3; each of these last two equations (with r non-negative) describes a ray, and together they describe the whole line.

3. Review and be ready to recognize and apply some standard trig. formulas. For example, today in class we used the formulas for double-angles:

sin (2x) = 2 (sin x)(cos x)

cos (2x) = (cos x)^2-(sin x)^2 = 1- 2 (sin x)^2 = 2 (cos x)^2 -1

4. Regarding using LH to verify that the cycloid has vertical tangents at theta = 0, 2 pi, 4 pi, etc.: the formula for the tangent slope is sin(theta)/(1-cos(theta)). When theta = 2 pi, for example, the ratio becomes 0/0. Thus, we take the limit of sin(theta)/(1-cos(theta)) as theta approaches 2 pi (from the left and separately from the right), apply LH to replace top and bottom by their derivatives: cos(theta)/sin(theta) (we still have the limit at this stage), and finally, we plug in theta = 2 pi to get 1/0 (0^+ if approaching 2 pi from the right, and 0^- if approaching 2 pi from the left). In any case, we get the tangent slope to be plus or minus infinity, which indicates a vertical tangent. Try to repeat the calculation from class that did NOT involve LH, but simplified first the tangent slope and then showed directly that where the tangent slope was infinity (or undefined).

5. The example r = 4 sin (theta): it is a single circle x^2 + (y-2)^2 = 4, as we concluded in class. It is not two circles on top of each other, although it is good to confirm this. In class we saw what happens when theta was between 0 and pi. When theta becomes larger than pi, e.g., theta = 5 pi/4, then r is negative: r = - 2 (square root(2)). When we make r positive, we need to subtract pi from theta, so we land on the point (r = 2 (square root(2)), theta = pi/4), which is one of the points on our circle that we already saw earlier. Thus, when theta becomes larger than pi, we start going over the same circle a second time, etc.

HW2A. Read: 10.2. Calculus with Parametric Curves (Review and finish). 10.4. Areas and Lengths in Polar Coordinates

Write Exercises:

10.2. #32,36,38,42,48,58 (in #38,58: just set up the integral), #62,66. Bonus: #73,74.

10.4. #4,6,8,12,18,28,32,38,46,48. Bonus: #36,44.

General notes on bonus HW problems: Bonus HW problems will be included in the weekly HW solutions. Ordinarily, they won't be included in quiz for that week, but they may be included in exams (or even later quizzes, if suitable), so you need to study their solutions carefully. Keep in mind that a "bonus" problem earlier in the semester may not look as hard later on in the semester.

HW2B. Read: 12.2.Vectors

Write Exercises:

12.2. #2,4,6,8,14,16,20,22,24,26,28,30,32,42,44,51. Bonus: #40; Prove that the three medians in a triangle intersect in one point (called the centroid of the triangle) and that this point divides each median in ratio 2:1 counted from the vertices of the triangle. (A median connects a vertex of a triangle with the midpoint of the opposite side.)

Note: Problem 10.2. #36(c) was assigned unintentionally. We will not be testing on exams about centroids of general figures. The only centroid for now that you should be aware of and be able to solve problems with is the centroid of a triangle (as we did in class). The first video below ("Triangle has a Magic Highway") and its extra footage will tell you more on that topic.

HW3A. Read: 12.3. The Dot Product (including Ex.7 on work)

Write Exercises:

12.3. #6-14,18,22-30,36,38,42,50,56,58*. Bonus: #61,62.

HW3B. Read: 12.4. The Cross Product. 12.5. Equations of Lines and Planes (only the first section on Lines, including Example 3 on p. 826)

Write Exercises:

12.4. #2-20,30,36,38,44. Bonus: #47,48.

12.5. #2,4,10,14,18,20,22. Bonus: #12.

HW4A. Read: 12.5. Equations of Lines and Planes (finish, concentrating on Planes and Distances, pp. 827-830). 12.6. Cylinders and Quadric Surfaces

Write Exercises:

12.5. #26,28,38,40,46,56,66,70,72,74,76 (in #76: use formula from #75). Bonus: #62,75.

12.6. #2,6,10,18,34,36,44,46,48. Bonus: #52.

HW4B. Read: 13.1. Vector Functions and Space Curves. 13.2. Derivatives and Integrals of Vector Functions

Write Exercises:

13.1. #2-12,16,18,28-32,42,44. Bonus: #46,48,50.

13.2. #2-10,18-22,26,28,34,36,40,42. Bonus: #16 (in two ways!),55.

General note on HW problems: Ordinarily, only even-numbered problems are assigned, unless an odd-numbered problem is explicitly included in HW. Thus, for example, #2-12 means #2,4,6,8,10.

HW5A. Read: 13.3. Arc Length and Curvature (up to Normal and Binormal Vectors (non-inclusive)).

Write Exercises:

13.3. #2,4,14,16,18,22,24,28,30,32,46. Bonus: #57 (only first sentence)

HW5B. Read: 13.3. Normal and Binormal Vectors (finish). 13.4. Motion in Space: Velocity and Acceleration (up to Kepler's Laws, non-inclusive). 14.1. Functions of Several Variables (up to Level Curves, non-inclusive)

Write Exercises:

13.3. #48,50,56. Bonus: #57 (finish)

13.4. #4,6,12,16,18(a),23,38,40,42. Bonus: #22,28,45

14.1. #2,10,14,16,18,20.

HW6. Read: 14.1. Functions of Several Variables (finish; concentrate on graphs,level curves, and 3-variable functions). 14.2. Limits and Continuity

Write Exercises:

14.1. #24,28,30,36,42,48,50,54,68,70,72.

14.2. #4,6,10,16,26,30,32,34,36,38*. Bonus: #40,41,44.

Notes on limit problems in HW6 and in general: When looking for the limit of z = f(x,y) at (a,b), there are several cases that can occur:

1. The function f(x,y) is continuous at (a,b) (use the CL's, Continuity Laws, and our conclusions about continuous functions from class). After stating this and why the function is continuous at (a,b), you can plug in (a,b) for (x,y) and the limit will come out easily. For example, 14.2 #6 in HW6 is about a rational function that is continuous at (2,-1). This type of situation is the easiest that can occur.

2. The function f(x,y) is not continuous at (a,b) because, for example, f(x,y) is not defined at (a,b).

(a) Try several different paths of approach to (a,b) and find the limits of f(x,y) along these paths. If two such limits are different, then conclude that the overall limit does not exist. See, for example, 14.2. #4,10,44 in HW6. This situation is of "intermediate" difficulty.

(b) Trying different paths of approach to (a,b) produces the same limit of f(x,y) along all these paths! This is the "toughest" situation. If you conjecture that the limit exists, you need to prove it, using other approaches. For example, substituting r = x^2+y^2 as in 14.2 #40,41 in HW6 simplifies the function and the limit to the single-variable case, and then you can appyl LH to the resulting single-variable function. If such a substitution does not exist, things get even more complicated. You might be able to apply the Squeeze Theorem as in 14.2 #16 in HW6. If nothing else works, the final resort is the epsilon-delta definition of limit.

In this course, for the time being, we will concentrate on cases #1 and #2(a), and perhaps, some simple cases of 2(b) where substitution works to reduce the problem directly to a single-variable function.

Two good examples on non-continuity, which you need to be able to do, involve piecewise-defined functions:

(A) f(x,y) = 2xy/(x^2+2y^2) for (x,y) not equal to (0,0) and f(0,0)=0 (or whatever else). This is like 14.2 #4 from HW6. The function is NOT continuous at (0,0) because the limit DNE at (0,0), regardless of what we define f(0,0) to be.

(B) f(x,y) = (x y^2+y^2 x)/(x^2-y^2) for (x,y) not equal to (2,-1) and f(2,-1)=1. This is like 14.2 #6 from HW6. The function is NOT continuous at (2,-1) because the limit at (2,-1) equals -2/3, while the value of f there is 1.

HW7A. Read: 14.3. Partial Derivatives (read on your own the example associated with Table 1 on p.912; read everything else except Partial DE on p.920 and Cobb-Douglas Production Function on p. 922 -- these are optional); 14.4. Tangent Planes and Linear Approximations

Write Exercises:

14.3. #2,6 (I find #6 tricky because one has to visualize and interpret what the picture is telling us and keep in mind the positive direction of all axes!), #12 (you can draw the graph by hand), #18,20,22,26,28,34,36,42,56,60,62; Bonus: #30,38,72

14.4. #4,6,12,14,16,17,18,19,20,21(don't graph); Bonus: #46

HW7B. Read: 14.4. Tangent Planes and Linear Approximations (finish; concentrate on differentiable functions: Definition 7 and Theorem 8; read also about Differentials for your general understanding); 14.5. Chain Rule. 14.6 Directional Derivatives(concentrate on directional derivatives up to Example 2, inclusive)

14.5. #1,2,4,6,8,10,14,16,20,22,28,30,34; Bonus: #49,51

14.6. #2 (read Example 1, p. 948), #4,6,8,12,14,20,28. Bonus: #30

HW8A. Read: 14.6. Directional Derivatives (starting from the Gradient Vector); 14.7. Maximum and Minimum Values (concentrate on critical points and how to find them using the gradient, and on Second Derivatives Test)

Write Exercises:

14.6. #21,22,23,24,29,34,38,42,44,50,54,56; Bonus: #52,62*

14.7. #2,4,6,8,12,14,16,18,20; Bonus: #22.

Notes on the Gradient and Its Applications from lecture on March 8. The following concepts and theorems were covered today in lecture:

1. Gradient

2. How gradient is used in computing directional derivatives

3. How gradient gives the answer to questions like: "In which direction is the rate of change of a function z = f(x,y) largest?"

4. How to locate the gradient on a contour map of z = f(x,y): the gradient points in the direction of largest ascent; the gradient is perpendicular to the tangent line to any level curve, thus the gradient can be thought of as a "normal" to any level curve.

5. Similarly, how to think of the gradient in regards to a level surface g(x,y,z) = c of a function t = g(x,y,z), the gradient of f points in the direction of largest ascent; the gradient is perpendicular to the tangent plane to the level surface, and thus the gradient can be thought of as a "normal" to the level surface.

6. The gradient as part of the equation of the tangent line to a level curve f(x,y) = c, and analogously, the gradient as part of the equation of the tangent plane to a level surface g(x,y,z) = c.

Warning about confusion between the formulas for tangent plane to a surface z = F(x,y) and for tangent plane to the level surface g(x,y,z) = c. The formulas eventually will give the same tangent plane equation, provided you use the formulas correctly. For example, if you have a surface z = F(x,y) and a point on the surface P(a,b,z=F(a,b)), you can use our previous formula for the tangent plane: z = F(a,b)+F_x(a,b)(x-a)+F_y(a,b)(y-b), OR you can turn your function into a level surface F(x,y)-z = 0 and treat this with the gradient approach for the function t = F(x,y)-z = g(x,y,z); the tangent plane will be g_x(P)(x-a)+g_y(P)(y-b)+g_z(P)(z-f(a,b)) = 0; since g_x = F_x and g_y = F_y (why?), while g_z = -1 (why?), we end up with exactly the same tangent plane equation as above: F_x(a,b)(x-a)+F_y(a,b)(y-b)-1(z-F(a,b)) = 0, or equivalently, solving for z we get z = F(a,b)+F_x(a,b)(x-a)+F_y(a,b)(y-b). Thus, yes, you can use either approach to find the equation of the tangent plane at a point on the graph of a FUNCTION z = F(x,y). But you have to be careful with the second approach, where you need to turn your function z = F(x,y) into the level surface of a "larger" function, namely, the level surface 0 = F(x,y)-z of the function t = F(x,y)-z. On the other hand, if you are already given a level surface (i.e., an equation of the form g(x,y,z) = c), you should proceed directly with the "gradient" approach, instead of trying to solve for z and use the "function" approach.

7. The gradient used to find critical points of functions z = f(x,y). Critical points yield ONLY POTENTIAL local min/max of a function.

8. And if the above is not enough, :), we also covered a theorem that is NOT using just the gradient, but something "more": the Second Derivatives Test, which yields the REALIZED (the actual) local min/max.

HW8B. Read: 14.7. Maximum and Minimum Values (finish; concentrate on finding global min/max). 14.8. Lagrange Multipliers (up to Two Constraints, non-inclusive)

Write Exercises:

14.7. #32-38,42-52; Bonus: #40 (do not graph),57

14.8. #4-12,15; Bonus: #14,16.

Notes on HW8B from 14.7: The exercises here fall into two major categories:

Type A: problems in which the function and the domain are already given (e.g., section 14.7, #32-38). The algorithm here goes as follows (assuming f is continuous on D, has continuous partial derivatives on D, and D is closed and bounded):

(1) find the critical points of f inside D (set gradient of f to be the zero vector); evaluate f(x,y) at these critical points.

(2) restrict f on the boundary of D; and each piece of this boundary, find the critical points of f (but now as a function of 1 variable); evaluate f at these points.

(3) evaluate f at the "endpoints" of each piece of the boundary.

(4) make a list of all these values of f, and determine which is the largest (the global max) and where it occurs; and which is the smallest (the global min) and where it occurs.

Type B: problems in which we need to first create the function from the situation in the problem (e.g., section 14.7 #42-52), determine the (desired) domain, and only then optimize the resulting function as in a Type A-problem. Typically, these are harder problems, but they fall into several subcategories: (minimal) distance (#42,44) problems, (maximal) area or volume problems (#48,50,52), "number" problems (#46). You can get better at these "word" problems only by doing more of them on your own, studying the HW solutions and redoing the harder-for-you problems again, coming regularly to lectures, and participating actively in sections.

Notes on HW8B from 14.8: The exercises here (section 14.8, #4-12) address ONLY the global min/max "along the boundary" (i.e., on a constraint curve or constraint surface). They are solved by setting by the system for Lagrange multipliers and solving it. We shall assume in these exercises that global min/max exist along the givent constraint. [In the next lecture we will finish the story of lagrange multipliers and how they give an alternative method for finding global min/max on closed+bounded domains and more.]

HW9A. Read: 15.1. Double Integrals over Rectangles. 14.8. Lagrange Multipliers (finish; Bonus: Two Constraints)

Write Exercises:

14.8. #22,23,24; Bonus: #18,20,49*.

15.1. #4-24,30,34-38,42,48; Bonus: #50,52.

HW9B. Read: 15.2. Double Integrals over General Regions (may skip for now Example 4 and all exercises on volumes in 15.2). 15.3. Double Integrals in Polar Coordinates (concentrate only on notes from lecture, i.e., writing Cartesian regions in polar coordinates)

Write Exercises:

15.2. #2-22,46-62; Bonus: #64,68.

15.3. #1,2,3,4,5,6.

HW10A. Read: 15.2. Double Integrals over General Regions (finish, concentrate on Example 4 and exercises on volumes in 15.2). 15.3. Double Integrals in Polar Coordinates (finish, read all examples). 15.4. (Optional) Applications of Double Integrals (concentrate only on Probability and Expected Values)

Write Exercises:

15.2. #24-30,36,38,66; Bonus: #32,69.

15.3. #8-26,30,32. Bonus: #38,40*.

(Optional, not to be included on exams:) 15.4. #27,28,29,30,33.

Notes on HW10A-10B: Material up to 15.3 and HW10A (inclusive) will be on Midterm 2.

HW10B. Read: 15.5. Surface Area. 15.6. Triple Integrals

Write Exercises:

15.5. #2,4,6,10,12,21,23; Bonus: #22*,24*.

15.6. #2,6,10,12,14,16,18,20,22,36,38; Bonus: 54,55(a)*.

HW11. Read: 15.8. Triple Integrals in Spherical Coordinates. 15.9. Change of Variables in Multiple Integrals. 16.1. Vector Fields (start reading as far as you can go in 16.1)

Write Exercises:

15.8. #2-12,18-26,30; Bonus: #28.

15.9. #2-10,15,16,17; Bonus: #12,14,18.

HW12A. Read: 16.1. Vector Fields. 16.2. Line Integrals, Part I (up to Line Integrals in Space, non-inclusive)

Write Exercises:

16.1. #2-18,22-26,30-32; Bonus: #34,36.

16.2. #2-8,19.

Notes on Lecture, 4/12/2016:

1. Section 16.1 on Vector Fields. We agreed to write a little vector on top of F for a vector field F(x,y) and F(x,y,z) since we cannot write in "boldface" as in the textbook. Please, adhere to this notation so that you are reminded at any moment whether you are working with scalar functions or vector functions/fields.

2. Part of Section 16.2. Line Integrals, as follows:

a. We defined for now informally the line integral of f(x,y) ds over a curve C as the area of a "shower curtain" hanging from a shower rod f|C (f restricted to C) all the way down to the curve C.
b. We have NOT yet defined the line integral using Riemann sums (Definition 2), nor have we proven the Formula 3 (p. 1075).
c. However, we introduced Formula 3 and we did a several examples with it.
d. We discussed the connection between ds and dt (as in the beginning of page 1076) via the arc-length formula.
e. We introduced a short-hand notation for Formula 3 as the integral from a to b of f(r(t))|r'(t)|dt where r(t) is the parametrizing vector function for C, and we recognized |r'(t)| as the speed along C.
f. We have NOT discussed yet that line integrals are generalization of ordinary integrals (third formula on page 1076).
g. We talked about splitting a curve C into pieces when calculating line integrals (bottom of page 1076).
h. We also introduced the special line integrals along C of f(x,y)dx and of f(x,y)dy by Formula 7 (page 1076), and talked about changing dx = x'(t)dt and dy=y'(t)dt.
i. We have NOT talked about these special integrals dx and dy via Riemann sums.

3. On Thursday:

a. We will cover all of the stuff for line integrals of scalar functions that was not covered on Tuesday (basically, everything that uses Riemann sums)
b. We will define and work with line integrals of vector fields.
c. We will interpret line integrals of vector fields as "work".
d. We will cover the statement and proof of the Fundamental Theorem for Line Integrals in 16.3.

HW12B. Read: 16.2. Line Integrals, Part II (finish). 16.3. Fundamental Theorem for Line Integrals (pp. 1087-1088)

Write Exercises:

16.2. #18-22;28,30,29(a),32(a),40-44; Bonus: #49,50; (Note: Just evaluate the integrals in #28,30. Do not use FTL Integrals in this unit).

16.3. #1,2; Bonus: Read as much as you can on your own in Section 16.3.

HW13A. Read: 16.3. Fundamental Theorem for Line Integrals (up to p. 1091, non-inclusive; optional: read section on "Conservation of Energy"). 16.4. Green's Theorem

Write Exercises:

16.3. #4-24, 26*; Bonus: #28,35.

16.4. #2-14,18,19; Bonus: #21.

HW13B. Read: 16.4. Green's Theorem (finish; especially the proof of GT). 16.3. Fundamental Theorem for Line Integrals (finish; Theorem 6 and pp. 1091-1093; Conservation of energy is optional). 16.5. Curl and Divergence (up to Example 5, inclusive)

Write Exercises:

16.3. #31,32,33,34; Bonus: #28,35.

16.4. #27,28,29; Bonus: #30 (redo without looking at your class notes).

16.5. #2-18.

HW14A. Read: 16.5. Curl and Divergence (finish; may skip for now Vector Forms of GT). 16.6. Parametric Surfaces and Their Areas (may skip "Surfaces of Revolution"). 16.7. Surface Integrals (of scalar functions; up to "Oriented Surfaces", non-inclusive).

Write Exercises:

16.5. #20,22,24,26,28,30; Bonus: #32. (Find in the text what "irrotational" and "incompressible" mean.)

16.6. #2-6, 14-26,34,40-48; Bonus: #50,64*(a)(c).

16.7. #6-20.

HW14B. Read: 16.7. Surface Integrals (of vector fields; finish). 16.8. Stokes' Theorem (may skip for now "Proof of a special case of ST" and "Meaning of curl F).

Write Exercises:

16.7. #22-32.

16.8. #2-10,12(a),14,16*,18*; Bonus: #17,19.

Solutions to HW14. The HW solutions will be taken off the web in a week and will not be distributed in any form later on, so make sure you download and study the HW Solution within the next few days.

HW15A (Optional). Read: 16.8 Stokes' Theorem (concentrate on Proof of ST in a special case, and Examples 1 and 2); 16.9. Divergence Theorem (Theorem Statement and Examples 1 and 2; may skip proof of DT)

Write Exercises (Optional):

16.9. #2-14,18,24.

Make your own summary tables, according to the given handout here Suggested Tables for a Review.

HW15B (Optional). Read: 16.10. Summary.

Write Exercises (Optional):

Review for Chapter 16: Concept Check. True-False Quiz: #1-13. Exercises: #3,9,10,11,13,15,17,19,21,25,27,29,33,35,36.

Videos on Game and Geometry Puzzles from the Numberphile Channel with Zvezdelina Stankova, directed by Brady Haran, containing some proof and problem-solving ideas that come handly in a variety of math areas:

Triangle has a Magic Highway (Euler Line): see how a triangle's center of mass relates to other centers of the triangle

Three Squares

Freedom for the Clones

Berkeley Math Circle