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Science and Math
Science and Math World

“What are Quotations?”

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Quotations are at once mundane and sublime. Whatever the philosophical stance, country, race, or religion of their origin; whether they be serious or whimsical; whether their creators are famous or notorious, controversial or celebrated, quotations are the essence of wisdom refined to a handful of well-chosen words.

They may inspire us to seek an understanding of their creators, to consider our own lives, to laugh, or merely to admire their mastery of language. However we use them, quotations summarize the collective insights of society, a legacy of knowledge passed onward, from one generation to the next.

**Like your body your mind also gets tired so refresh it by wise sayings.**

– Ali bin Abi Talib

(source)

Cross-country skiing, also known as Nordic skiing, is a part of many longstanding disciplines in the Winter Games. Unlike downhill skiing, cross-country events typically involve mass starts, which allow athletes to interact, using one another for pacing and tactics. Drafting can be a valuable method to save energy and reduce drag. A following skier sees a 25% drag reduction while drafting; the lead skier gets about a 3% reduction in drag compared to skiing solo. Competitors usually wear tight-fitting suits to minimize drag, but unlike speedskating, for example, cross-country skiers don’t get much benefit from roughened surfaces and impermeable fabrics. Typical race speeds are 4 – 9 m/s, and most of these high-tech fabrics don’t provide tangible benefits until higher speeds. (Image credit: Reuters/S. Karpukhin, US Biathlon, GettyImages/Q. Rooney)

**This is a picture of 100 of solutions.**

In each, a quarter of the cross is coloured in red. You can explore a prezi of them here. A high-resolution picture can be downloaded here (PNG 4MB) and the original SVG file here.

Ideas used to construct my solutions by *David Butler.*

- Fraction addition and subtraction.
- Halving retangles and triangles using symmetry.
- Adding a bit to a shape and taking the same bit don’t change the area.
- Moving the apex of a triangle parallel to its base doesn’t change the area.
- Shearing a shape doesn’t change its area.
- Making a shape half as high and twice as wide doesn’t change the area.
- Shrinking all the lengths by half makes a shape with a quarter the area.
- The formula for the area of a trapezium.
- The formula for the area of a crossed trapezium.
- The Lune of Hippocrates.
- Pythagoras’ theorem.
- Rotational symmetry.
- The sum of a geometric series.