Theorem of the Day^{®} 
Proudly hosting Carnival of Mathematics 160 


Today's Theorem on Tuesday 14 August, 2018
Welcome to a gallery whose exhibits are the crowning achievements of mathematics: her theorems. Each day offers a different theorem (or lemma, law, formula or identity), each one worthy of adorning the walls of a mathematical Abattoirs, Baltic, Cairn House, Guggenheim, Louvre, Nail Factory, Staatliche Museen, Tate, Uffizi or Zach Feuer. Each theorem has been presented so as to be appreciated by as wide an audience as possible. If the statement of the theorem appears obscure, pass on to the illustration and its accompanying explanation. If it still seems hard it is probably because it is hard. But no more than O'Keeffe's Blue and Green Music, say, is a 'hard' painting or Hepworth's Two Figures is a 'hard' sculpture. It is there for you to engage with on your own terms. By 'engage' I mean: admire it, turn it over in your mind, try to follow the example, if one is given; if you are studying it online, follow the web link, which will provide a pictorial interpretation, a proof or even a clever animation. Click on the 'further reading' link. It will usually take you to the amazon.co.uk page for the book. This is a convenient way to provide full bibliographical details of the book and often, a preview of its contents; it is not a recommendation that you buy the book (but, if you do, this website earns a small referral fee). Each theorem is as selfcontained as possible; when viewing a theorem online, the button, topright, links to a Glossary page where there is a brief explanation of some parts of mathematical language; an accompanying indicates that the theorem has been provided with some supplementary information often by expert visitors to the page. At the bottom of the page, the arrow will take you to a related but usually less sophisticated theorem which may shed light on today's; the button will take you to the complete theorem list; and the arrow may point to a theorem which leads on from today's or takes you deeper into its subject. For some theorems are deeper than others; some use more technical language than others; some are just harder to understand than others. Some days are harder than others. I hope even the most difficult days offer something of wonder. 
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