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mathworld.wolfram.com 28/09/2015

mathworld.wolfram.com

The curve given by the polar equation

r=a(1-costheta),
(1)
sometimes also written

r=2b(1-costheta),
(2)
where b=a/2.

The cardioid has Cartesian equation

(x^2+y^2+ax)^2=a^2(x^2+y^2),
(3)
and the parametric equations

x = acost(1-cost)
(4)
y = asint(1-cost).
(5)
The cardioid is a degenerate case of the limaçon. It is also a 1-cusped epicycloid (with r=r) and is the catacaustic formed by rays originating at a point on the circumference of a circle and reflected by the circle.

The cardioid has a cusp at the origin.

The name cardioid was first used by de Castillon in Philosophical Transactions of the Royal Society in 1741. Its arc length was found by la Hire in 1708. There are exactly three parallel tangents to the cardioid with any given gradient. Also, the tangents at the ends of any chord through the cusp point are at right angles. The length of any chord through the cusp point is 2a

mathworld.wolfram.com

[09/28/15]   Unsolved Problems
There are many unsolved problems in mathematics. Some prominent outstanding unsolved problems (as well as some which are not necessarily so well known) include

1. The Goldbach conjecture.

2. The Riemann hypothesis.

3. The conjecture that there exists a Hadamard matrix for every positive multiple of 4.

4. The twin prime conjecture (i.e., the conjecture that there are an infinite number of twin primes).

5. Determination of whether NP-problems are actually P-problems.

6. The Collatz problem.

7. Proof that the 196-algorithm does not terminate when applied to the number 196.

8. Proof that 10 is a solitary number.

9. Finding a formula for the probability that two elements chosen at random generate the symmetric group S_n.

10. Solving the happy end problem for arbitrary n.

11. Finding an Euler brick whose space diagonal is also an integer.

12. Proving which numbers can be represented as a sum of three or four (positive or negative) cubic numbers.

13. Lehmer's Mahler measure problem and Lehmer's totient problem on the existence of composite numbers n such that phi(n)|(n-1), where phi(n) is the totient function.

14. Determining if the Euler-Mascheroni constant is irrational.

15. Deriving an analytic form for the square site percolation threshold.

16. Determining if any odd perfect numbers exist.

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