# CTK Insights

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29 Feb

### Jewels in the Bride's Chair

In particular, the are three points (the diagram shows only one) where four straight lines meet forming successive angles of 45°. The curious thing about this is that the observation that has been made just a few years ago is a very direct, nay, immediate consequence of the properties that have been known for more than 100 years. This is certainly a remarkable feature of Vecten's configuration having four concurrent lines spaced evenly; the fact holds for any triangle. Of the four lines, two are equal and in the remaining pair one is twice as long as the other.

26 Feb

### Right and ... left

TweetEvery parent is likely to have had an experience I once read about. A father placed four sweets on a table and asked his 4 year old daughter to count. She did this perfectly: one, two, three, four. On his prompting she also observed that there were 4 sweets. The father suggested she pick one […]

24 Feb

### Cartesian axes and parallel lines

I always feel apprehension on seeing the number line or x-axis introduced as a horizontal line, with the numbers increasing left to right. When the time comes, the y-axis is commonly introduced as perpendicular to the x-axis and hence vertical. Seldom there is enough time for a trifle observation that the direction of a number line is irrelevant and that the choice of the orientation of the system of axes in the plane is a matter of convenience. The trouble comes when there is a need to change variables or, say, introduce the inverse function.

15 Feb

### Dividing circular area into equal parts

TweetIt's not simple and sometimes impossible to divide a circle into n equal parts. That is, if the word "circle" signifies the border of a circular region, then, as was shown by the great Gauss, the construction with compass and ruler is only possible when Euler function φ is a power of 2: φ(n) = […]

11 Feb

### Reformists, pay attention

TweetI strongly believe that in a few years the thrust of an education reform will converge on the web. The process has begun even if not yet sunctioned by powers that be at the US Congress. The best universities put up their courses online, see, for example, a New York Times article. The general accesibility of […]

11 Feb

### Our standards for school graduation

TweetQuiz yourself: how old is that quote: Our standard for high school graduation has slipped badly. Fifty years ago a high-school diploma meant something. We have simply misled our students and misled the nation by handing out high-school diplomas to those who we well know had none of the intellectual qualifications that a high-school diploma […]

11 Feb

### The case for a reform

TweetQuiz yourself. Would you know where the quote below comes from? It is to be hoped that the near future will bring reforms in the mathematical teaching in this country. We are in sad need of them. From nearly all of our colleges and universities comes the loud complaint of inefficient preparation on the part […]