Type of Spirals: A spiral is a curve in the plane or in the space, which runs around a centre in a special way.
Different spirals follow. Most of them are produced by formulas:The radius r(t) and the angle t are proportional for the simplest spiral, the spiral of Archimedes. Therefore the equation is:
(3) Polar equation: r(t) = at [a is constant].
From this follows
(2) Parameter form: x(t) = at cos(t), y(t) = at sin(t),
(1) Central equation: x²+y² = a²[arc tan (y/x)]².
You can make a spiral by two motions of a point: There is a uniform motion in a fixed direction and a motion in a circle with constant speed. Both motions start at the same point.
(1) The uniform motion on the left moves a point to the right. - There are nine snapshots.
(2) The motion with a constant angular velocity moves the point on a spiral at the same time. - There is a point every 8th turn.
(3) A spiral as a curve comes, if you draw the point at every turn(Image).
Figure 1: (1) Archimedean spiral - (2) Equiangular Spiral (Logarithmic Spiral, Bernoulli’s Spiral).
Figure 2 : (1) Clothoide (Cornu Spiral) - (2) Golden spiral (Fibonacci number).
More Spirals: If you replace the term r(t)=at of the Archimedean spiral by other terms, you get a number of new spirals. There are six spirals, which you can describe with the functions f(x)=x^a [a=2,1/2,-1/2,-1] and f(x)=exp(x), f(x)=ln(x). You distinguish two groups depending on how the parameter t grows from 0.
Figure 4: If the absolute modulus of a function r(t) is increasing, the spirals run from inside to outside and go above all limits. The spiral 1 is called parabolic spiral or Fermat’s spiral.
Figure 5: If the absolute modulus of a function r(t) is decreasing, the spirals run from outside to inside. They generally run to the centre, but they don’t reach it. There is a pole. Spiral 2 is called the Lituus (crooked staff).
Figure 7: Spirals Made of Line Segments.
Source: Spirals by Jürgen Köller.
See more on Wikipedia: Spiral, Archimedean spiral, Cornu spiral, Fermat’s spiral, Hyperbolic spiral, Lituus, Logarithmic spiral,
Fibonacci spiral, Golden spiral, Rhumb line, Ulam spiral,
Hermann Heights Monument, Hermannsdenkmal.
MATH MYTHS: (from Mind over Math)
1. MEN ARE BETTER IN MATH THAN WOMEN.
Research has failed to show any difference between men and women in mathematical ability. Men are reluctant to admit they have problems so they express difficulty with math by saying, “I could do it if I tried.” Women are often too ready to admit inadequacy and say, “I just can’t do math.”
2. MATH REQUIRES LOGIC, NOT INTUITION.
Few people are aware that intuition is the cornerstone of doing math and solving problems. Mathematicians always think intuitively first. Everyone has mathematical intuition; they just have not learned to use or trust it. It is amazing how often the first idea you come up with turns out to be correct.
3. MATH IS NOT CREATIVE.
Creativity is as central to mathematics as it is to art, literature, and music. The act of creation involves diametrical opposites—working intensely and relaxing, the frustration of failure and elation of discovery, satisfaction of seeing all the pieces fit together. It requires imagination, intellect, intuition, and aesthetic about the rightness of things.
4. YOU MUST ALWAYS KNOW HOW YOU GOT THE ANSWER.
Getting the answer to a problem and knowing how the answer was derived are independent processes. If you are consistently right, then you know how to do the problem. There is no need to explain it.
5. THERE IS A BEST WAY TO DO MATH PROBLEMS.
A math problem may be solved by a variety of methods which express individuality and originality-but there is no best way. New and interesting techniques for doing all levels of mathematics, from arithmetic to calculus, have been discovered by students. The way math is done is very individual and personal and the best method is the one which you feel most comfortable.
6. IT’S ALWAYS IMPORTANT TO GET THE ANSWER EXACTLY RIGHT.
The ability to obtain approximate answer is often more important than getting exact answers. Feeling about the importance of the answer often are a reversion to early school years when arithmetic was taught as a feeling that you were “good” when you got the right answer and “bad” when you did not.
7. IT’S BAD TO COUNT ON YOUR FINGERS.
There is nothing wrong with counting on fingers as an aid to doing arithmetic. Counting on fingers actually indicates an understanding of arithmetic-more understanding than if everything were memorized.
8. MATHEMATICIANS DO PROBLEMS QUICKLY, IN THEIR HEADS.
Solving new problems or learning new material is always difficult and time consuming. The only problems mathematicians do quickly are those they have solved before. Speed is not a measure of ability. It is the result of experience and practice.
9. MATH REQUIRES A GOOD MEMORY.
Knowing math means that concepts make sense to you and rules and formulas seem natural. This kind of knowledge cannot be gained through rote memorization.
10. MATH IS DONE BY WORKING INTENSELY UNTIL THE PROBLEM IS SOLVED. Solving problems requires both resting and working intensely. Going away from a problem and later returning to it allows your mind time to assimilate ideas and develop new ones. Often, upon coming back to a problem a new insight is experienced which unlocks the solution.
11. SOME PEOPLE HAVE A “MATH MIND” AND SOME DON’T.
Belief in myths about how math is done leads to a complete lack of self-confidence. But it is self-confidence that is one of the most important determining factors in mathematical performance. We have yet to encounter anyone who could not attain his or her goals once the emotional blocks were removed.
12. THERE IS A MAGIC KEY TO DOING MATH.
There is no formula, rule, or general guideline which will suddenly unlock the mysteries of math. If there is a key to doing math, it is in overcoming anxiety about the subject and in using the same skills you use to do everything else.
Source: “Mind Over Math,” McGraw-Hill Book Company, pp. 30-43.
Revised: Summer 1999
Student Learning Assistance Center (SLAC)
Southwest Texas State University
Another myth: Arithmetic is math. It’s one little bit and lots of mathematicians hate arithmetic.
everything here is good except #4. there is certainly a need to know how you got your answer, or what your answer means, or else what you’re doing isn’t actually math. granted, it can sometimes be easier to get an answer than to explain an answer, but the explanation is always more important than the answer itself (this is true in mathematics, at least. in other disciplines, take your answer and run)
One of applications of “slope” to explain puzzles and paradoxes -
Triangle Dissection Paradox
"Below the two parts moved around - The partilisions are exactly the same, as those used above - From where "come" this hole?"
Explain: In the figure, the slope of the “hypotenuse” in figure 1 and figure 2 are completely different. (Click on image to see full size).
Also, The above two figures are rearrangements of each other, with the corresponding triangles and polyominoes having the same areas. Nevertheless, the bottom figure has an area one unit larger than the top figure (as indicated by the grid square containing the dot).The source of this apparent paradox is that the “hypotenuse” of the overall “triangle” is not a straight line, but consists of two broken segments. As a result, the “hypotenuse” of the top figure is slightly bent in, whereas the “hypotenuse” of the bottom figure is slightly bent out. The difference in the areas of these figures is then exactly the “extra” one unit. Explicitly, the area of triangular “hole” (0, 0), (8, 3), (13, 5) in the top figure is 1/2, as is the area of triangular “excess” (0, 0), (5, 2), (13, 5) in the bottom figure, for a total of one unit difference. Source: Triangle Dissection Paradox on Mathworld.wolfram.
- Slope: In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m. Slope is calculated by finding the ratio of the “vertical change” to the “horizontal change” between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient (“rise over run”), giving the same number for every two distinct points on the same line. The rise of a road between two points is the difference between the altitude of the road at those two points, say y1 and y2, or in other words, the rise is (y2 − y1) = Δy. For relatively short distances - where the earth’s curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (x2 − x1) = Δx. Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line.
- Also, here are the direction of a line is either increasing, decreasing, horizontal or vertical:
- A line is increasing if it goes up from left to right. The slope is positive, i.e. m>0.
- A line is decreasing if it goes down from left to right. The slope is negative, i.e. m<0.
- If a line is horizontal the slope is zero. This is a constant function.
- If a line is vertical the slope is undefined (see below).
The steepness, incline, or grade of a line is measured by the absolute value of the slope. A slope with a greater absolute value indicates a steeper line - Source: Slope on Wikipedia