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enHilbert's hotel
https://plus.maths.org/content/hilberts-hotel
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/keys_icon2.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Suppose you're a hotel manager and your hotel is full. That's great, of course, but there's always the temptation to squeeze in more guests. In real life, this might mean clearing out the broom cupboards and getting bad reviews on Trip Advisor, but in the world of maths it's no problem. As long as your hotel has infinitely many rooms, that is.</p>
<div class="rightimage" style="max-width: 300px;"><img src="/content/sites/plus.maths.org/files/articles/2017/Hilbert/Hotel.jpg" alt="Hotel" width="300" height="400" />
<p>Welcome to Hilbert's hotel!</p>
</div>
<!-- Image MF -->
<p>The idea goes back to the German mathematician <a href="http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Hilbert.html">David Hilbert</a>, who used the example of a hotel to demonstrate the counter-intuitive games you can play with infinity. Suppose that your hotel has infinitely many rooms, numbered 1, 2, 3, etc. All rooms are occupied, when a new guest arrives and asks to be put up. What do you do? It's easy. Ask the guest in room 1 to move to room 2, the guest in room 2 to move into room 3, the guest in room 3 to move into room 4, and so on. If there were only finitely many rooms, the guest in the last room would have nowhere to go, but since there are infinitely many, everybody will find a new abode. You'll have to ask the guests to move simultaneously though, because if you ask them to move one after the other, the move might take an infinite amount of time, since infinitely many guests have to move. </p>
<p>Using this trick you can actually accommodate any finite number of new guests. If <img src="/MI/8e2f6b74f2802c94f7ebb40f5cc0e8cf/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> new guests arrive, simply ask each existing guest to move to the room whose number is <img src="/MI/8e2f6b74f2802c94f7ebb40f5cc0e8cf/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> plus the number of their existing room. As an example, if there are <img src="/MI/8e2f6b74f2802c94f7ebb40f5cc0e8cf/images/img-0002.png" alt="$8$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> new guests, then the guest currently in room <img src="/MI/8e2f6b74f2802c94f7ebb40f5cc0e8cf/images/img-0003.png" alt="$10$" style="vertical-align:0px;
width:15px;
height:12px" class="math gen" /> needs to move into room <img src="/MI/8e2f6b74f2802c94f7ebb40f5cc0e8cf/images/img-0004.png" alt="$10+8=18$" style="vertical-align:-1px;
width:82px;
height:13px" class="math gen" />. </p>
<p>But things get better still. Suppose an infinite number of new guests arrive, forming an orderly queue outside the hotel. In this case, ask each existing guest to move into the room whose number is twice the number of their current room. So, a guest staying in room <img src="/MI/a662b6e3496d24f0dfba47b4b8214231/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> moves to room <img src="/MI/a662b6e3496d24f0dfba47b4b8214231/images/img-0002.png" alt="$2x$" style="vertical-align:0px;
width:17px;
height:12px" class="math gen" />. After this manoeuvre only the even numbered rooms are occupied: rooms <img src="/MI/a662b6e3496d24f0dfba47b4b8214231/images/img-0003.png" alt="$2 = 2 \times 1$" style="vertical-align:0px;
width:66px;
height:12px" class="math gen" />, <img src="/MI/a662b6e3496d24f0dfba47b4b8214231/images/img-0004.png" alt="$4 = 2 \times 2$" style="vertical-align:0px;
width:67px;
height:12px" class="math gen" />, <img src="/MI/a662b6e3496d24f0dfba47b4b8214231/images/img-0005.png" alt="$6 = 2 \times 3$" style="vertical-align:0px;
width:67px;
height:12px" class="math gen" />, and so on. The odd numbered rooms are all free, so you can put your first new guest into room 1, the second new guest into room 3, the third new guest into room 5, and so on. Everybody is happy. </p>
<h3>Infinite layering</h3>
<p>This isn’t all. Suppose an infinite number of coaches arrive, each carrying an infinite number of new guests. Assume, for simplicity, that the coaches are numbered 1, 2, 3, etc, and that the seats in each coach are also numbered 1, 2, 3, etc. You start by asking each existing guest to move into the room whose number is twice the number of their current room, as before. This leaves the odd numbered rooms free again. Now tell the passenger of coach 1 with seat 1 to move into room 3, the passenger of coach 1 with seat 2 into room <img src="/MI/0ad529bcc65f2a6cb4944fbbd1a3f130/images/img-0001.png" alt="$3^2 = 9,$" style="vertical-align:-3px;
width:50px;
height:17px" class="math gen" /> the passenger of coach 1 with seat 3 into room <img src="/MI/0ad529bcc65f2a6cb4944fbbd1a3f130/images/img-0002.png" alt="$3^3 = 27$" style="vertical-align:0px;
width:54px;
height:14px" class="math gen" />, and so on. In other words, the passenger of coach 1 with seat <img src="/MI/0ad529bcc65f2a6cb4944fbbd1a3f130/images/img-0003.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> moves into room <img src="/MI/0ad529bcc65f2a6cb4944fbbd1a3f130/images/img-0004.png" alt="$3^ n$" style="vertical-align:0px;
width:16px;
height:12px" class="math gen" />. Any power of <img src="/MI/0ad529bcc65f2a6cb4944fbbd1a3f130/images/img-0005.png" alt="$3$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> is odd, so all these rooms are guaranteed to be free. </p>
<div class="leftimage" style="max-width: 424px;"><img src="/content/sites/plus.maths.org/files/articles/2017/Hilbert/keys.jpg" alt="Hotel keys" width="424" height="283" />
<p>There's room for everyone.</p>
</div>
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<p>What about people arriving in the second coach? Here’s how to accommodate them: the guest with seat 1 in coach 2 moves into room 5, the guest with seat 2 in coach 2 moves into room <img src="/MI/1c07d0f7063a7e03b50c814934b1eb46/images/img-0001.png" alt="$5^2 = 25$" style="vertical-align:0px;
width:54px;
height:14px" class="math gen" />, the guest with seat 3 in coach 3 moves into room <img src="/MI/1c07d0f7063a7e03b50c814934b1eb46/images/img-0002.png" alt="$5^3 = 125$" style="vertical-align:0px;
width:62px;
height:14px" class="math gen" />, and so on. In other words, the guest with seat <img src="/MI/1c07d0f7063a7e03b50c814934b1eb46/images/img-0003.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> in coach 2 moves into room <img src="/MI/1c07d0f7063a7e03b50c814934b1eb46/images/img-0004.png" alt="$5^ n.$" style="vertical-align:0px;
width:21px;
height:13px" class="math gen" /> </p>
<p>How do you continue this for the third coach? Well, 3 and 5 are consecutive prime numbers. The next prime number along is 7, so we put the passenger of coach 3 with seat number <img src="/MI/634e0f864d97cf73c2de6b89444a4639/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> into room <img src="/MI/634e0f864d97cf73c2de6b89444a4639/images/img-0002.png" alt="$7^ n$" style="vertical-align:0px;
width:15px;
height:13px" class="math gen" />. The next prime number after 7 is 11, so we put the passenger of coach 4 with seat number <img src="/MI/634e0f864d97cf73c2de6b89444a4639/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> into the room <img src="/MI/634e0f864d97cf73c2de6b89444a4639/images/img-0003.png" alt="$11^ n$" style="vertical-align:0px;
width:23px;
height:12px" class="math gen" />. Generally, you put the passenger of coach <img src="/MI/634e0f864d97cf73c2de6b89444a4639/images/img-0004.png" alt="$m$" style="vertical-align:0px;
width:14px;
height:7px" class="math gen" /> with seat number <img src="/MI/634e0f864d97cf73c2de6b89444a4639/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> into the room <img src="/MI/634e0f864d97cf73c2de6b89444a4639/images/img-0005.png" alt="$p^ n$" style="vertical-align:-3px;
width:17px;
height:14px" class="math gen" />, where <img src="/MI/634e0f864d97cf73c2de6b89444a4639/images/img-0006.png" alt="$p$" style="vertical-align:-3px;
width:10px;
height:10px" class="math gen" /> is the <img src="/MI/634e0f864d97cf73c2de6b89444a4639/images/img-0007.png" alt="$(m+1)st$" style="vertical-align:-4px;
width:68px;
height:18px" class="math gen" /> prime number. </p>
<p>Are all these rooms guaranteed to be free? The answer is yes. All the room numbers of the new guests are powers of prime numbers. There's a beautiful result known as the <a href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic"><em>fundamental theorem of arithmetic</em></a>, which says that every whole number can be written as a product of primes in a unique way.<p> This means that if room number <img src="/MI/7683f363e07cf9e45f04e86238a11174/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> is a power of some prime, then it can’t be a power of another prime. For example, we know that guest number 5 from coach number 1 is allocated room <img src="/MI/7683f363e07cf9e45f04e86238a11174/images/img-0002.png" alt="$3^5 = 243.$" style="vertical-align:0px;
width:66px;
height:14px" class="math gen" /> Now if there were another guest allocated to the same room, say guest 3 from coach 2, then <img src="/MI/7683f363e07cf9e45f04e86238a11174/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> would also have to be a power of another prime, eg <img src="/MI/7683f363e07cf9e45f04e86238a11174/images/img-0003.png" alt="$x=5^3$" style="vertical-align:0px;
width:45px;
height:14px" class="math gen" />, which, according to the fundamental theorem it can’t be (and isn’t since <img src="/MI/7683f363e07cf9e45f04e86238a11174/images/img-0004.png" alt="$5^3=125$" style="vertical-align:0px;
width:62px;
height:14px" class="math gen" />). </p></p>
<div style="max-width: 340px; float: right; border: thin solid grey;
background: #CCC CFF; padding: 0.5em; margin-left: 1em; font-size:
75%">
<h3>Proof for three layers of infinity</h3><p>Suppose a guest arrives on ship <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0001.png" alt="$s_1$" style="vertical-align:-2px;
width:14px;
height:9px" class="math gen" />, coach <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0002.png" alt="$m_1$" style="vertical-align:-2px;
width:21px;
height:9px" class="math gen" /> and seat <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0003.png" alt="$n_1$" style="vertical-align:-2px;
width:16px;
height:9px" class="math gen" />. This means that she’ll go into room </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0004.png" alt="\[ q_1^{p_1^{n_1}}, \]" style="width:34px;
height:24px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> where <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0005.png" alt="$p_1$" style="vertical-align:-3px;
width:15px;
height:10px" class="math gen" /> is the <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0006.png" alt="$(m_1+1)st$" style="vertical-align:-4px;
width:75px;
height:18px" class="math gen" /> prime number and <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0007.png" alt="$q_1$" style="vertical-align:-3px;
width:13px;
height:10px" class="math gen" /> is the <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0008.png" alt="$(s_1+1)st$" style="vertical-align:-4px;
width:69px;
height:18px" class="math gen" /> prime number. Now suppose another guest, on ship <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0009.png" alt="$s_2$" style="vertical-align:-2px;
width:14px;
height:9px" class="math gen" />, coach <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0010.png" alt="$m_2$" style="vertical-align:-2px;
width:21px;
height:9px" class="math gen" /> and seat <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0011.png" alt="$n_2$" style="vertical-align:-2px;
width:16px;
height:9px" class="math gen" />, is allocated to the same room. Then </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0012.png" alt="\[ q_1^{p_1^{n_1}} = q_2^{p_2^{n_2}}, \]" style="width:87px;
height:24px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> where <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0013.png" alt="$p_2$" style="vertical-align:-3px;
width:15px;
height:10px" class="math gen" /> is the <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0014.png" alt="$(m_2+1)st$" style="vertical-align:-4px;
width:75px;
height:18px" class="math gen" /> prime number and <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0015.png" alt="$q_2$" style="vertical-align:-3px;
width:13px;
height:10px" class="math gen" /> is the <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0016.png" alt="$(s_2+1)st$" style="vertical-align:-4px;
width:69px;
height:18px" class="math gen" /> prime number. </p><p>By the fundamental theorem of arithmetic, this means that <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0017.png" alt="$q_1 = q_2$" style="vertical-align:-3px;
width:50px;
height:10px" class="math gen" /> and <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0018.png" alt="$p_1^{n_1} = p_2^{n_2}$" style="vertical-align:-4px;
width:68px;
height:17px" class="math gen" />. Applying the fundamental theorem again shows that the latter implies that <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0019.png" alt="$p_1 = p_2$" style="vertical-align:-3px;
width:53px;
height:10px" class="math gen" /> and <img src="/MI/b3124a821cf8c3edbe402e1add6b7392/images/img-0020.png" alt="$n_1 = n_2.$" style="vertical-align:-2px;
width:61px;
height:9px" class="math gen" /> Hence the second guest is actually the same as the first. </p></div>
<p>We’ll carry on for one more level. Suppose an infinite number of ships arrive, each carrying an infinite number of coaches, each carrying an infinite number of guests. Each guest now comes with three numbers attached to them: the number of their ship, the number of their coach and the number of their seat in the coach. You start by moving all the existing guests to the even-numbered rooms as before. Now suppose a guest arrived on ship <img src="/MI/30dda4bb7195be9559c870a38187fd85/images/img-0001.png" alt="$s$" style="vertical-align:0px;
width:7px;
height:7px" class="math gen" />, coach <img src="/MI/30dda4bb7195be9559c870a38187fd85/images/img-0002.png" alt="$m$" style="vertical-align:0px;
width:14px;
height:7px" class="math gen" /> and seat <img src="/MI/30dda4bb7195be9559c870a38187fd85/images/img-0003.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" />. We start as before: find the <img src="/MI/30dda4bb7195be9559c870a38187fd85/images/img-0004.png" alt="$(m+1)st$" style="vertical-align:-4px;
width:68px;
height:18px" class="math gen" /> prime number, call it <img src="/MI/30dda4bb7195be9559c870a38187fd85/images/img-0005.png" alt="$p$" style="vertical-align:-3px;
width:10px;
height:10px" class="math gen" />, and raise it to the power of the seat number. That gives <img src="/MI/30dda4bb7195be9559c870a38187fd85/images/img-0006.png" alt="$p^ n$" style="vertical-align:-3px;
width:17px;
height:14px" class="math gen" />. Now for the ship number, find the <img src="/MI/30dda4bb7195be9559c870a38187fd85/images/img-0007.png" alt="$(s+1)st$" style="vertical-align:-4px;
width:61px;
height:18px" class="math gen" /> prime number, call it <img src="/MI/30dda4bb7195be9559c870a38187fd85/images/img-0008.png" alt="$q$" style="vertical-align:-3px;
width:8px;
height:10px" class="math gen" />, and raise that to the number you already have, to get </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/30dda4bb7195be9559c870a38187fd85/images/img-0009.png" alt="\[ q^{p^ n}. \]" style="width:28px;
height:19px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>As an example, if the passenger arrived on ship 1, coach 1 and had seat number 2, then they should move into room </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/30dda4bb7195be9559c870a38187fd85/images/img-0010.png" alt="\[ 3^{3^2} = 3^{9} = 19683. \]" style="width:127px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> Numbers get very large very quickly with this approach. The guest from ship <img src="/MI/30dda4bb7195be9559c870a38187fd85/images/img-0011.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> and coach <img src="/MI/30dda4bb7195be9559c870a38187fd85/images/img-0012.png" alt="$3$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> with seat number <img src="/MI/30dda4bb7195be9559c870a38187fd85/images/img-0013.png" alt="$4$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> should go into room </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/30dda4bb7195be9559c870a38187fd85/images/img-0014.png" alt="\[ 5^{7^4}, \]" style="width:26px;
height:21px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> but if you try to work out that number, you’ll find that your calculator will give up. In theory, though, it’s possible to accommodate all the guests (another appeal to the fundamental theorem of arithmetic will prove this, see the box) and since this is maths, theory is all that counts. </p>
<p>So far we have managed to accommodate guests that arrive in three layers of infinity: infinitely many ships, each carrying infinitely many coaches, each carrying infinitely many guests. Can we go further? The answer is yes. You could accommodate guests arriving in any finite number of layers of infinity. What you can't always do, however, is accommodate guests arriving in infinitely many layers of infinity. But then, that's never going to happen anyway.</p>
<p><em>You can find out more about infinity on our <a href="/content/infinity-page">What is infinity?</a> page.</em></p>
<hr/>
<h3>About the author</h3>
<p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>. </p></div></div></div>Mon, 13 Feb 2017 15:00:21 +0000mf3446786 at https://plus.maths.org/contenthttps://plus.maths.org/content/hilberts-hotel#commentsMonkeys play the peanut lottery
https://plus.maths.org/content/monkeys-play-peanut-lottery
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/monkey_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Maths is useful for everyone, not just humans. To be successful in your
hunting and
foraging, and to keep track of your family members and
predators, you need to be able to gauge quantities, which is why numerical ability developed early on in
evolution. Many animals, including some fish and insects, use some
sort of understanding of quantities to navigate their lives.</p>
<div class="rightimage" style="max-width: 321px;"><img src="/content/sites/plus.maths.org/files/news/2017/monkeys/monkey.jpg" alt="A capuchin monkey." width="321" height="292"/>
<p>A capuchin monkey. </p>
</div>
<!-- Image from fotolia.com -->
<p>That's a very basic type of maths, but what about something a
little more sophisticated, such as, for example,
probabilities? A <a href="http://link.springer.com/article/10.1007/s10071-016-1043-9">recent study</a>
with capuchin monkeys addressed this question, and seems to suggest that, at least
some of them, have an
intuitive grasp of chances.</p>
<p>To understand probabilities it's not enough to understand absolute
quantities. Suppose I show you a jar containing 100 peanuts and 100
pebbles, and another jar containing 7 peanuts and 3 pebbles. If all
you understood were absolute quantities, and I gave you the chance to
pick one item from one of the jars, you'd go for the first jar: many
more peanuts in that one. However, the chance of picking a peanut is
much higher for the second jar, and that's the one you should have
chosen. What's important here are <em>proportions</em> (eg 100
out of 200 versus 7 out of 10) and understanding how they link to the
act of randomly picking an item from a bigger collection.</p>
<p>Such an understanding seems a lot harder than basic counting, and scientists used to think that children don't develop
it until they are about seven. Recent
studies have shown, however, that even 12-month-old babies have basic
probabilistic abilities. The same goes for all four species of great
apes: orangutans, gorillas, bonobos and chimpanzees. </p>
<p>Monkeys are the obvious next step when it comes to seeing whether
probabilistic abilites are shared by animals further removed from
humans. Capuchin monkeys are particularly interesting because, being
<em><a href="https://en.wikipedia.org/wiki/New_World_monkey">new world monkeys</a></em>, their genetic lineage split off from that of
humans over 30 million years ago. If a genetic trait that's
shared by humans and capuchins comes from a common ancestor, then
it must be very, very old.</p>
<div style="float:left; margin-right: 1em;"><iframe width="400" height="225" src="https://www.youtube.com/embed/Ovay0eepIEU" frameborder="0" allowfullscreen></iframe>
<p style="width: 400px; font-size: small; color: purple;">A monkey making its choice. (The monkeys that took part in the study did't have to live in these little cubicles, they only came into them for the chance of a treat.) Video from Springer, Animal Cognition, <em>Intuitive Probabilistic Inference in Capuchin Monkeys</em>, 2016, Tecwyn, E.C., Denison, S., Messer, E.J.E. et al., with permission of Springer.</p>
</div>
<p>To test the monkeys' probabilistic finesse, the researcher set up a
peanut lottery similar to the example above. Monkeys were shown
two glass jars containing a mixture of peanuts, which they like a lot,
and monkey pellets, which they don't like as much. A researcher then
shook each jar and pretended to pick one item from each (in fact she
already had an item concealed in each hand but the monkeys didn't know
that). To convice the monkeys that the choice was random, she closed
her eyes and tilted her head upwards while she was doing the
picking. She then held her closed fists out for the monkey to pick
one. Some of the time she crossed her arms over first, to check if the
monkeys were really picking a hand, rather than just the side that
corresponded to their preferred jar.</p>
<p>Some of the monkeys clearly didn't bother with thinking about
probabilities: they always picked the the same side, no
matter what. Among the others, however, the researchers claim to have
detected a trend to go for the hand with the
better chance of a peanut. This, they say, is evidence that at least
some capuchin monkeys intuitively understand proportions and what they
mean for the chance of getting a peanut. </p>
<p>This is the first time that a
"capacity
for rudimentary intuitive statistics" appears to have been found in monkeys. If
the capuchin monkeys really do possess it, then this might mean that
the ancestor they share with humans did too, which would mean that the
trait is ancient. It could
of course be that the two lineages developed the capacity
independently. More studies will be needed to
find out.</p></div></div></div>Fri, 10 Feb 2017 14:36:12 +0000mf3446785 at https://plus.maths.org/contenthttps://plus.maths.org/content/monkeys-play-peanut-lottery#commentsMaths in a minute: The golden ratio
https://plus.maths.org/content/maths-minute-golden-ratio
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/golden_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>The golden ratio was first studied by the ancient Greeks, and it
was defined in terms of geometry. Take a piece of line and divide it
into a longer and a shorter piece so that the ratio between the whole
piece and the longer piece is the same as the ratio between the longer
piece and the shorter piece. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/packages/2017/Fibonacci/line.png" alt="Double slit" width="289" height="63" />
<p style="max-width: 289px;">Divide the line so that the ratio between the whole line (<em>a+b</em>) and the longer piece (<em>a</em>) is equal to the ratio between the longer piece (<em>a</em>) and the shorter piece (<em>b</em>).</p>
</div>
<!-- Image made by MF -->
<p>Let’s write <img src="/MI/7e4145c0fb311212cbd7eea6d68ce03d/images/img-0001.png" alt="$\phi $" style="vertical-align:-3px;
width:10px;
height:14px" class="math gen" /> for that ratio. We can work out the value of <img src="/MI/7e4145c0fb311212cbd7eea6d68ce03d/images/img-0001.png" alt="$\phi $" style="vertical-align:-3px;
width:10px;
height:14px" class="math gen" /> quite easily. Suppose that after the division the longer piece of line has length <img src="/MI/7e4145c0fb311212cbd7eea6d68ce03d/images/img-0002.png" alt="$a$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> and the shorter one has length <img src="/MI/7e4145c0fb311212cbd7eea6d68ce03d/images/img-0003.png" alt="$b$" style="vertical-align:0px;
width:7px;
height:11px" class="math gen" />. Then, since <img src="/MI/7e4145c0fb311212cbd7eea6d68ce03d/images/img-0001.png" alt="$\phi $" style="vertical-align:-3px;
width:10px;
height:14px" class="math gen" /> equals the ratio between whole line and <img src="/MI/7e4145c0fb311212cbd7eea6d68ce03d/images/img-0002.png" alt="$a$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> as well as the ratio between <img src="/MI/7e4145c0fb311212cbd7eea6d68ce03d/images/img-0002.png" alt="$a$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> and <img src="/MI/7e4145c0fb311212cbd7eea6d68ce03d/images/img-0003.png" alt="$b$" style="vertical-align:0px;
width:7px;
height:11px" class="math gen" />, we get </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/7e4145c0fb311212cbd7eea6d68ce03d/images/img-0004.png" alt="\[ \frac{a+b}{a} = \frac{a}{b} = \phi . \]" style="width:109px;
height:33px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>The left hand side of the equation can be written as </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/7e4145c0fb311212cbd7eea6d68ce03d/images/img-0005.png" alt="\[ 1+\frac{b}{a}, \]" style="width:44px;
height:33px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> and since </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/7e4145c0fb311212cbd7eea6d68ce03d/images/img-0006.png" alt="\[ \frac{b}{a} =\frac{1}{\frac{a}{b}} = \frac{1}{\phi }, \]" style="width:88px;
height:40px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>we have </p><table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/7e4145c0fb311212cbd7eea6d68ce03d/images/img-0007.png" alt="\[ 1+ \frac{1}{\phi } = \phi . \]" style="width:78px;
height:38px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>Multiplying through by <img src="/MI/7e4145c0fb311212cbd7eea6d68ce03d/images/img-0001.png" alt="$\phi $" style="vertical-align:-3px;
width:10px;
height:14px" class="math gen" /> gives </p><table id="a0000000006" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/7e4145c0fb311212cbd7eea6d68ce03d/images/img-0008.png" alt="\[ \phi + 1 = \phi ^2. \]" style="width:82px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>Using the quadratic formula to solve this equation we get a positive root </p><table id="a0000000007" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/7e4145c0fb311212cbd7eea6d68ce03d/images/img-0009.png" alt="\[ \phi = \frac{1+\sqrt {5}}{2} = 1.61803.... \]" style="width:181px;
height:37px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>The golden ratio has all sorts of interesting properties, but perhaps the most famous one is its link to the
<a href="/content/fibonacci-sequence-brief-introduction">Fibonacci
sequence</a>.</p>
<p>This famous sequence of numbers starts with two 1s. Subsequent
numbers are formed by adding the two previous ones. This gives</p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/da5fb43e790118bf34dda5811329c58d/images/img-0001.png" alt="\[ 1,1,2,3,5,8,13,21,34,55,89,.... \]" style="width:232px;
height:16px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>The sequence can be continued indefinitely. You can form a new
sequence by dividing each number in the Fibonacci sequence by the
number that came before:</p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/9f551994997185e6b517442e1d20a725/images/img-0001.png" alt="\[ 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55, .... \]" style="width:415px;
height:17px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>Here's a picture of this sequence:</p>
<div class="centreimage"><img src="/issue3/fibonacci/graph.gif" alt="1/1, 2/1, 3/2, 5/3, 8/5,..." width="394" height="261" />
<p>Ratio of successive Fibonacci terms.</p>
</div>
<p>You can see that, as you move down the sequence, the numbers
approach a limiting value. And that limiting value is exactly the
golden ratio!</p>
<p>To find out more about the Fibonacci sequence and the golden ratio,
including their appearance in nature and supposed link to aesthetics, see <a href="/content/fantastic-fibonacci">this selection of
articles</a>.</p></div></div></div>Thu, 09 Feb 2017 16:54:53 +0000mf3446788 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-golden-ratio#commentsFantastic Fibonacci
https://plus.maths.org/content/fantastic-fibonacci
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/rabbits_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/packages/2017/Fibonacci/rabbits.jpg" alt="Rabbits" width="350" height="244" />
<p></p>
</div>
<!-- Image fromm fotolia -->
<p> Bees do it, rabbits do it, and luckily, we humans can do it too: explore the famous Fibonacci sequence</p>
<p>1, 1, 2, 3, 5, 8, 13, 21, ...</p>
<p>First discovered in the 12th century by Leonardo Fibonacci while thinking about a problem involving baby rabbits, the sequence has since been found all over nature, from the spirals in sunflowers to the family tree of bees. It comes with a number called the <em>golden ratio</em>, famous for its interesting mathematical properties and a (supposed) connection to our sense of beauty.
</p>
<p>The following articles will tell you all you need to know about the Fibonacci sequence and the golden ratio. Enjoy!</p>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/5/19_dec_2014_-_1020/fib_icon.jpg" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/fibonacci-sequence-brief-introduction">The Fibonacci sequence: A brief introduction</a> </strong> — Find out how the famous sequence arose from a problem involving rabbits.
</p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/4/4_nov_2013_-_1159/icon.jpg" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/life-and-numbers-fibonacci">The life and numbers of Fibonacci</a> </strong> — See how the sequence also appears in the family trees of bees, in the turns of sea shells and sunflower seed heads, and how it all stemmed from a simple example in one of the most important books in Western mathematics.
</p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/golden_icon.jpg" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/maths-minute-golden-ratio">Maths in a minute: The golden ratio</a> </strong> — A quick introduction to the famous number and how it is linked to the Fibonacci sequence.
</p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/packages/2017/Fibonacci/Ratio1_icon.jpg" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/golden-ratio-and-aesthetics">The golden ratio and aesthetics</a> </strong> — It was Euclid who first defined the golden ratio, and ever since people have been fascinated by its extraordinary properties. Find out if beauty is in the eye of the beholder, and how the golden ratio crosses from mathematics to the arts.
</p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/packages/2017/Fibonacci/phic.jpg" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/maths-minute-continued-fractions">Maths in a minute: Continued fractions</a> </strong> — A quick introduction to why the golden ratio is the most irrational number of them all.
</p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/packages/2017/Fibonacci/Saturn_small.jpg" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/chaos-numberland-secret-life-continued-fractions">Chaos in Numberland: The secret life of continued fractions</a> </strong> — A deeper look at continued fractions, which provide the most revealing representations of many numbers, and what they have to do with the golden ratio.
</p></div></div></div></div>Thu, 09 Feb 2017 15:06:26 +0000mf3446787 at https://plus.maths.org/contenthttps://plus.maths.org/content/fantastic-fibonacci#commentsDo we live in a hologram?
https://plus.maths.org/content/do-we-live-hologram
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/hologram_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="max-width: 306px;"><img src="/content/sites/plus.maths.org/files/news/2017/hologram/hologram2.jpg" alt="Hologram" width="306" height="291"/>
<p>
</p>
</div>
<p>Physicists claim they have <a href="http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.041301">discovered evidence</a> that the Universe might be a hologram. They have compared the idea with information about the
Universe that comes from the <em>cosmic microwave background</em> (the left-over radiation from
the Big Bang) and concluded that it may be as plausible as the
accepted theory describing the Universe.</p>
<p>In ordinary life holograms are images that appear three-dimensional,
but are created, through clever tricks, by a
two-dimensional surface. When it comes to the Universe, the idea is that all the
physics that happens in the world is, in a sense, generated by
happenings on a two-dimensional boundary surface. "Imagine that
everything you see, feel and hear in three dimensions (and your
perception of time) in fact emanates from a flat two-dimensional
field," says Kostas Skenderis of the University of Southampton, one
of the authors of the new research. "The idea is similar to that of ordinary holograms where a three-dimensional image is encoded in a two-dimensional surface, such as in the hologram on a credit card. However, this time, the entire Universe is encoded!"</p>
<p>The idea becomes a little easier to grasp if you forget about
visual holograms and look at the picture below. It's a disc with lots
of lines drawn in. To describe the picture, you don't really need to
describe each line in its entirety: you only need to say what its
end-points are. Thus, the entire disc universe is described by pairs
of points on its boundary. Even if the lines were to move around, the
information on the pairs of points is sufficient to describe their
motion. In a similar, though hugely more
complicated way, physicists have suggested that our Universe has a
boundary, and that to describe stuff going on inside it, you only need
to describe what's happening on that boundary.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/news/2017/hologram/disc.png" alt="Impact" width="256" height="235"/>
<p style="max-width: 256px;">Each line is defined by its two end points on the boundary circle of the disc.
</p>
</div>
<p>It's a strange idea, but it didn't come from out of nowhere. Its
origins lie with those
gravitational monstrosities we call black holes. A black hole is
formed when a lot of mass is concentrated in a tiny
region of space. If there's enough mass, then its gravitational pull
will be
so strong that nothing, not even light, can escape from the vicinity
of the black hole
— hence the name. Black holes come with a boundary of no return
called their <em>event horizon</em>: once you pass that boundary
you'll be gobbled up by the black hole, never to return.</p>
<p>Physicists initially thought of black holes as very simple objects, until a
particular problem started to bug them. Were you to fall into a black hole then,
naively, it would seem that your fall leaves
the world a simpler place: you're a complex being,
and all
that complexity would have disappeared forever. Such loss of
complexity, of information, is not permitted by the laws of physics,
however. It violates something called the <em><a href="/content/maths-minute-second-law-thermodynamics">second law of
thermodynamics</a></em>. (You can test that law out for yourself by not
bothering to tidy your bedroom for a while. Left undisturbed it will
soon disintegrate into chaos, that is, greater complexity containing
more information.)
So physicists eventually came around to the idea
that black holes aren't just voids, but can be
thought of as containing information.</p>
<p> But how much information can be contained in a black hole? If you
think of information as having a physical manifestation (which it
always does), for example in the pages of a book, then the question should
amount to asking how much stuff you can fit into the black
hole. Obviously, that amount of stuff should be measured by the
volume of the black hole. It turns out, however, that the information
content (to be precise, the <em>entropy</em>) of a black hole is measured by the surface area of its event horizon (see <a
href="/content/bekenstein">this <em>Plus</em> article</a> to find out more). </p>
<p>This discovery, made in the 1970s, was the first step towards
holography: it suggested that the information content of a black hole
is something fundamentally two-dimensional. Physicists later went on
to suggest that the same goes for ordinary regions of space.
Like with
our disc universe above, whatever is going on inside them can
essentially be encoded in a two-dimensional boundary surface. If this
is the case, then our three-dimensional description of physics
overshoots the mark, and we should be able to get by with a leaner
version.</p>
<div class="leftimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2017/hologram/HologramIllustration_small.jpg" alt="Hologram" width="350" height="302"/>
<p>In a holographic universe you can describe the action of
dropping an apple in the interior purely in terms of the boundary
theory.
</p>
</div>
<!-- Image created by Charles Trevelyan -->
<p>In 1997 developments in this area inspired <a href="https://www.ias.edu/scholars/maldacena">Juan
Maldacena</a>, only 29 at the time, to create a toy universe
(described by mathematics) that is entirely determined by a theory defined on
its boundary. Like the real Universe, this toy universe contains
fundamental particles that can be described mathematically. But each
of those fundamental particles has its counterpart on the boundary,
also described mathematically, and every interaction
between interior particles corresponds exactly to an interaction
between boundary particles. You can now describe the action of, say,
dropping an apple in the interior purely in terms of the boundary
theory. You can even choose to ignore the interior altogether without
losing any information at all — in this sense Maldacena's toy world is a
hologram. (You can read more about this toy world and its development
in <a href="/content/illusory-universe">this <em>Plus</em> article</a> based on an
interview with Maldacena.)</p>
<p>So far, so similar to our disc universe above. But there's an
added, and very important, attraction. The boundary of Maldacena's world is described by a
<em>quantum field theory</em> — that's the kind of theory used to
describe the smallest building blocks of matter, such as atoms and
sub-atomic particles, and their interaction through the electromagnetic
force and the strong and weak nuclear forces. Crucially, quantum field
theories can't describe the force of gravity. This failing is one of
the biggest problems of modern physics and it makes Maldacena's
invention all the more amazing: even though it is generated by a gravity-less quantum field theory defined on the
boundary, the physics in the interior of his
universe <em>does</em> include the force of gravity (for those
familiar with the area, it
is described by a <em>string theory</em>). Thus, in Maldacena's
universe, gravity emerges
holographically! So the work may contain a clue as to how quantum physics
and the theory describing gravity, Einstein's general relativity, may
finally be reconciled.
</p>
<p>Maldacena's work described a world very different from
our own, but since then physicists have come up
with theories that could, potentially, apply to the Universe we
actually live in.
It's these kind of models that were tested in the
new research. The team behind it, Niayesh Afshordi, Claudio Corianò, Luigi Delle Rose, Elizabeth Gould and Kostas Skenderis, worked out what holographic models of the very early Universe would imply for the structure of the
cosmic microwave background. They then compared these kind of predictions
with the real cosmic microwave background and found that a
holographic universe could indeed be possible.</p>
<p> "Holography is a huge leap forward in the way we think about the
structure and creation of the Universe," says Skenderis. "Scientists
have been working for decades to combine Einstein's theory of gravity
and quantum theory. Some believe the concept of a holographic universe
has the potential to reconcile the two. I hope our research takes us
another step towards this."</p></div></div></div>Wed, 08 Feb 2017 09:50:44 +0000mf3446784 at https://plus.maths.org/contenthttps://plus.maths.org/content/do-we-live-hologram#commentsPhysics in a minute: The double slit experiment
https://plus.maths.org/content/physics-minute-double-slit-experiment-0
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/interference_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>One of the most famous experiments in physics is the double
slit experiment. It demonstrates, with unparalleled strangeness, that
little particles of matter have something of a wave about them, and suggests that the very act of
observing a particle has a dramatic effect on its behaviour.</p>
<p>To start off, imagine a wall with two
slits in it. Imagine throwing tennis balls at the wall. Some will
bounce off the wall, but some will travel through the slits. If there's
another wall behind the first, the tennis balls that have travelled
through the slits will hit it. If you mark all the spots where a ball
has hit the second wall, what do you expect to see? That's right. Two
strips of marks roughly the same shape as the slits.</p>
<p>In the image below, the first wall is shown from the top, and the
second wall is shown from the front.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/double_balls.png" alt="Double slit" width="350" height="303" />
<p style="max-width: 350px;">The pattern you get from particles.</p>
</div>
<!-- Image made by MF -->
<p>Now imagine shining a light (of a single colour, that is, of a
single wavelength) at a wall with two slits (where the distance between the slits is roughly the same as the light's wavelength). In the image below, we show the
light wave and the wall from the top. The blue lines
represent the peaks of the wave. As the wave passes though both
slits, it essentially splits into two new waves, each spreading out from one of the slits. These two waves then interfere with each other. At some points, where a peak meets a trough, they will cancel each other out. And at
others, where peak meets peak (that's where the blue curves cross in the diagram), they will reinforce each other. Places where the waves reinforce each other give the brightest
light. When the light meets a second wall placed behind the first, you will
see a stripy pattern, called an <em>interference pattern</em>. The bright
stripes come from the waves reinforcing each other.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/double_waves.png" alt="Double slit" width="350" height="323" />
<p style="max-width: 350px;">An interference pattern.</p>
</div>
<!-- Image made by MF -->
<p>Here is a picture of a real interference pattern. There are more
stripes because the picture captures more detail than our diagram. (For the sake of correctness, we should say that the image also shows a <em>diffraction pattern</em>, which you would get from a single slit, but we won't go into this here, and you don't need to think about it.)</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/interference.jpg" alt="Double slit" width="558" height="240" />
<p style="max-width: 558px;">Image: <a href="https://commons.wikimedia.org/wiki/File:Single_slit_and_double_slit2.jpg">Jordgette</a>, <a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA 3.0</a>.</p>
</div>
<p>Now let's go into the quantum realm. Imagine firing electrons at
our wall with the two slits, but block one of those slits off for the moment. You'll find that some of the electrons will pass through the open slit and strike the second wall just as tennis balls would: the spots they arrive at form a strip roughly the same shape as the slit.</p>
<p>Now open the second slit. You'd expect two rectangular strips on the second wall, as with the tennis balls, but what you actually see is very different: the spots where electrons hit build up to replicate the
interference pattern from a wave.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/electrons.png" alt="Double slit" width="350" height="298" />
<p style="max-width: 350px;"></p>
</div>
<!-- Image made by MF -->
<p>Here is an image of a real double slit experiment with electrons. The individual pictures show the pattern you get on the second wall as more and more electrons are fired. The result is a stripy interference pattern.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/electrons2.jpg" alt="Double slit" width="350" height="205" />
<p style="max-width: 350px;">Image: <a href="https://commons.wikimedia.org/wiki/File:Double-slit_experiment_results_Tanamura_four.jpg">Dr. Tonomura and Belsazar</a>, <a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA 3.0</a></p>
</div>
<p> How can this be?</p>
<p>One possibility might be that the electrons somehow interfere with each other, so they don't arrive in the same places they would if they were alone. However, the interference pattern
remains even when you fire the electrons one by one, so that they have
no chance of interfering. Strangely, each individual electron contributes one dot to an overall pattern that looks like the interference pattern of a wave. </p>
<p> Could it
be that each electrons somehow splits, passes through both slits at once,
interferes with itself, and then recombines to meet the second screen as a single, localised particle?</p>
<p>To find out, you might place a detector by the slits, to see which
slit an electron passes through. And that's the really weird bit. If
you do that, then the pattern on the detector screen turns into the
particle pattern of two strips, as seen in the first picture above! The interference pattern disappears. Somehow, the very act of
looking makes sure that the electrons travel like well-behaved
little tennis balls. It's as if they knew they were being spied on and decided not to be caught in the act of performing weird quantum shenanigans.</p>
<p>What does the experiment tell us? It suggests that what we call "particles", such as electrons, somehow combine characteristics of particles and characteristics of waves. That's the famous <em>wave particle duality</em> of quantum mechanics. It also suggests that the act of observing, of measuring, a quantum system has a profound effect on the system. The question of exactly how that happens constitutes the <em>measurement problem</em> of quantum mechanics.</p>
<hr><h3>Further reading</h3>
<ul><li>For an extremely gentle introduction to some of the strange aspects of quantum mechanics, read <a href="/content/watch-and-learn"><em>Watch and learn</em></a>. </li>
<li>For a gentle introduction to quantum mechanics, read <a href="/content/ridiculously-brief-introduction-quantum-mechanics"><em>A ridiculously short introduction to some very basic quantum mechanics</em></a>.</li>
<li>For a more detailed, but still reasonably gentle, introduction to quantum mechanics, read <a href="/content/schrodinger-1"><em>Schrödinger's equation — what is it?</em></li></div></div></div>Sun, 05 Feb 2017 12:37:46 +0000mf3446774 at https://plus.maths.org/contenthttps://plus.maths.org/content/physics-minute-double-slit-experiment-0#commentsMaths in a minute: Penguins
https://plus.maths.org/content/maths-minute-penguins
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/penguin_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="ax-width: 300px;"><img src="/content/sites/plus.maths.org/files/blog/012017/penguin.jpg" alt="Penguin" width="300" height="400" />
<p>Why do penguins rotate their eggs?</p>
</div>
<!-- Image fromm fotolia -->
<p>Today, January 20th, is penguin awareness day. We can't let this important day pass without looking at some relevant maths. Penguins are well-known, and respected, for the incredible hardships some species endure in order to raise their chicks, including gruelling walks across Antarctic ice from nesting to feeding grounds. One of the lesser-known, and less challenging, parental duties of a penguin is to regularly rotate its unhatched eggs about their long axis. </p><p>For a very long time nobody knew why they do it, but it was important to find out: captive breeding programmes of penguins usually take the eggs away from the incubating bird and place them in an incubator. The mother will then lay more eggs to make up for the loss, so as long as the artificial incubators do their job well, there will be more chicks than the mother alone could produce. To be maximally efficient, the artificial incubators need to mimic natural conditions as closely as possible, including the rotation of the eggs. But without knowing why the eggs are being rotated it's hard to replicate the rotation effectively, and without the rotations the chicks won't hatch.</p>
<p>This is where maths comes in. Back in 2003 Bristol Zoo asked a study group of mathematicians at the <a href="https://www.ktn-uk.co.uk">Knowledge Transfer Network</a> to investigate the penguin rotation problem by looking at the fluid dynamics within the egg. An initial idea was that the rotation ensures a uniform temperature of 37°C throughout the egg, but the study group's mathematical model, based on the equations of fluid dynamics, showed that the effects of rotation on temperature are minimal. They had more luck with a second theory: their model showed that the rotations produced the necessary mixing within the egg, ensuring that nutrients are distributed uniformly and waste is dispersed. The model described the turning and churning within the egg in detail and provided a theoretical basis to help with programming artificial incubators, not only for penguins but also for other species of bird. The group also recommended what type of further research would be necessary to improve their model. </p>
<p>It's safe to assume, therefore, that there are currently penguins alive on this planet that owe their existence directly to an application of fluid dynamics. Who would have thought! The Industrial Mathematics Knowledge Transfer Network aims to help businesses benefit from academic expertise in mathematics. You can find out more about its work, including on penguins, in <a href="/content/career-interview-business-development-manager-smith-institute">this <em>Plus</em> article</a>.</p></div></div></div>Fri, 20 Jan 2017 14:06:25 +0000mf3446773 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-penguins#commentsWhat made the Moon?
https://plus.maths.org/content/what-made-moon
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/moon_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Few things in life are as dependable as the Moon, yet the origin of this familiar companion is still shrouded in mystery.
According to the <a href="https://en.wikipedia.org/wiki/Giant-impact_hypothesis">leading theory</a>, the Moon formed from the debris of a catastrophic collision between Earth and another body roughly the size of Mars. A new study, however, suggests another scenario: perhaps the Moon we see today isn't the Earth's first, but made up of several <em>moonlets</em> that formed from smaller collisions and eventually merged into one.</p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2016/Moon/impact_impact.jpg" alt="Impact" width="350" height="280"/>
<p>This artist's concept shows celestial bodies colliding. A similar collision could have produced our Moon. Image: <a href="https://www.nasa.gov/multimedia/imagegallery/image_feature_1454.html">NASA/JPL-Caltech</a>. </p>
</div>
<p>
Theories of the formation of the Moon are based on mathematics. If you know the correct equations you can calculate, or nowadays simulate on a computer, how different scenarios could have given birth to the Moon. It was
Charles Darwin's son, <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Darwin.html">George Howard Darwin</a>, who came up with one such theory back in the late nineteenth century. According to Darwin, the Moon formed from molten material that was spun off the Earth by centrifugal forces as it turned around its own axis, aided by tidal forces coming from the Sun. Darwin used Newton's physics to calculate that, in the past, the Moon did indeed orbit much closer to the Earth, and that it is slowly moving further away from it now.
He was right on that account — experiments performed by the Apollo 11 mission confirmed the Moon's drifting away from the Earth — but he couldn't prove that the Moon had indeed been part of the Earth at one point.</p>
<p> The idea that the Moon was formed by a single catastrophic impact, rather than a spinning away of Earth material, started to find acceptance in the 1970s. Today
proponents of the <em>giant impact</em> or <em>Big Splash</em> theory believe that the collision happened around 4.5 billion years ago. The body which collided with the Earth is fittingly called <a href="https://en.wikipedia.org/wiki/Theia">Theia</a> after the Greek titan and mother of the Moon goddess <a href="https://en.wikipedia.org/wiki/Selene">Selene</a>.</p>
<p>But while there's good <a href="https://en.wikipedia.org/wiki/Giant-impact_hypothesis">evidence to support the giant impact theory</a>, there are also problems.
In the 1980s computer simulations of the impact showed that the Moon would have been formed mostly from remnants of Theia, and should therefore have a chemical make-up very different from that of Earth: no other body in our solar system matches the Earth in its composition, so in all likelihood Theia didn't either. But the opposite is actually the case.
Analyses of Moon rocks show that the Moon's composition matches that of the Earth's <em><a href="https://en.wikipedia.org/wiki/Mantle_(geology)">mantle</a></em>, the layer that lies between its crust and its core. It seems, then, that the Moon was formed from Earth material, rather than the stuff of Theia, just as Darwin had suggested.</p>
<p>Scientists weren't ready to give up on the giant impact theory through. They have shown that
more Earth material would have been flung into space if the early Earth was spinning much faster than it does today, or if Theia was larger than had previously been assumed, about the size of the Earth itself. In both cases a Moon as we know it today could have been formed, but both would have produced an Earth-Moon system spinning at a faster rate than the one we now observe. There are theories for what might have slowed the system down, but not everyone is convinced. What is more, other simulations have suggested that the system's current spin (and mass) might be the result of more than one impact.</p>
<div class="leftimage" style="max-width: 243px;"><img src="/content/sites/plus.maths.org/files/news/2016/Moon/moon.jpg" alt="Moon" width="243" height="240"/>
<p>The Moon tinted reddish during a lunar eclipse. Image: <a href="https://commons.wikimedia.org/wiki/File:Lunar_eclipse_October_8_2014_California_Alfredo_Garcia_Jr_mideclipse.JPG">Alfredo Garcia</a>, <a href="https://creativecommons.org/licenses/by-sa/2.0/deed.en">CC BY-SA 2.0</a>. </p>
</div>
<p>It's this multiple impact idea that the new research picks up on. <a href="http://physics.technion.ac.il/~hperets/">Hagai Perets</a>, Raluca Rufo and <a href="https://www.psi.edu/about/staffpage/aharonson">Oded Aharonson</a>
ran 800 simulations to investigate possible impact scenarios and what they could mean for Moon formation. "Our model suggests that the ancient Earth once hosted a series of moons, each one formed from a different collision with the proto-Earth," <a href="https://www.sciencedaily.com/releases/2017/01/170109125125.htm">says Perets</a>. "It's likely that such moonlets were later ejected, or collided with Earth or with each other to form bigger moons."</p>
<p>The tidal forces of the Earth would have caused the small moons to move away from it and the gravitational influence of these moons on each other could have caused them to change their motion around the Earth. "It's likely that small moons formed through the process could cross orbits, collide and merge," says Rufo. "A long series of such moon-moon collisions could gradually build up a bigger moon — the Moon we see today." The researchers believe that this scenario, because it involves a slow evolution rather than a single collision under very specific conditions, is also more likely than a single giant impact. </p>
<p>The research was <a href="http://www.nature.com/ngeo/journal/vaop/ncurrent/full/ngeo2866.html">published</a> in <em>Nature Geoscience</em> last week, and we'll have to wait and see how the multiple impact idea will be received by other Moon researchers. In the mean time the Moon will keep slowly moving away from Earth at a <a href="http://www.bbc.co.uk/news/science-environment-12311119">rate of around 3.8cm a year</a>. But that's only as fast as a finger nail grows, so in our life times and beyond, the Moon will remain a steady companion.</p></div></div></div>Wed, 18 Jan 2017 17:46:03 +0000mf3446771 at https://plus.maths.org/contenthttps://plus.maths.org/content/what-made-moon#commentsMaths in a minute: The graph isomorphism problem
https://plus.maths.org/content/maths-minute-graph-isomorphism-problem
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/map-icon.png" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Networks are everywhere. We encounter them every day, in the form of the internet, or the social, transport and utility networks we depend upon. Mathematically, a network is an example of a <em>graph</em>: a collection of nodes (also called <em>vertices</em>) connected up by links (also called <em>edges</em>). The problem is that the same graph can appear in different guises — which prompts the question of how hard it is to tell whether two apparently different graphs are actually the same. That question is known as the <em>graph isomorphism problem</em>.</p>
<div style="float: right;"><table><tr><td><div class="rightimage" style="width: 300px;"><img src="/sites/plus.maths.org/files/news/2015/graphiso/map3.png" alt="Accurate London Underground map" width="300" height="254" />
<p></p>
</div><!-- Image from wikipedia, creative commons but no author named -->
</td></tr><tr><td>
<div class="rightimage" style="width:300px;"><img src="/content/sites/plus.maths.org/files/news/2015/graphiso/tube-1.jpg" width="300" height="199" alt="Tube map"/><p> A piece of a geographically accurate map of the London Underground (top, image from <a href="https://commons.wikimedia.org/wiki/File:London_Underground_Zone_1.svg">Wikimedia commons</a>) and the (roughly) corresponding piece of the network on the actual map (bottom).</p></div><!-- Image from istockphoto, first appeared here https://plus.maths.org/content/happy-birthday-london-underground --></td></tr></table></div>
<p>An example of two graphs that look different but are actually the same comes from the London Underground map. The relative locations of the stations have been changed, compared to the geographical map, to make it easier to read (find out more <a href="https://plus.maths.org/content/happy-birthday-london-underground">here</a>). The Underground network on the tube map looks different from what it looks like when drawn accurately, yet the two are isomorphic: you can match each vertex and each edge from one graph exactly to a vertex or edge in the other (and vice versa), in a way that preserves the connectivity of the graph (which vertex is linked to which). In this example we know from the start that the two networks are isomorphic, after all they represent the same physical train network, but without the context it might have taken us a while to figure out that they really are.</p>
<p>The graph isomorphism problem doesn't ask <em>whether</em> it's possible spot if two graphs, given by different representations, are isomorphic. There are algorithms, step-by-step recipes, which can do just that. Rather, the question is whether there is an algorithm that is faster than the ones that are known. Mathematicians have a way of ranking problems according to their difficulty (more precisely, their complexity). The graph isomorphism problem asks where in the hierarchy of complexity classes the graph question belongs. Although an answer wouldn't have many practical uses (people have long known of algorithms that efficiently solve the problem for the vast majority of graphs they come across in the real world), it's a biggie in complexity theory and theoretical computer science. You can find out more about the graph isomorphism problem in <a href="https://plus.maths.org/content/middle-class-problems?nl=0"><em>Middle class problems</em></a> and about complexity classes in <a href="https://plus.maths.org/content/not-just-matter-time-part-1?nl=0"><em>Not just a matter of time</em></a>.</p>
</div></div></div>Wed, 18 Jan 2017 11:26:01 +0000mf3446772 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-graph-isomorphism-problem#commentsThe festival of the spoken nerd
https://plus.maths.org/content/node/6770
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>In our ongoing research into the hypothesis that maths and science are the new rock'n'roll we ventured to the seedy streets of Soho, to see <em><a href="http://festivalofthespokennerd.com">Festival of the spoken nerd</a></em> at Soho Theatre. Otherwise known as <a href="http://stevemould.com">Steve Mould</a>, <a href="http://helenarney.com">Helen Arney</a> and <a href="http://standupmaths.com">Matt Parker</a>, the FotSN have been touring their combination of comedy, science and maths for a while and are now filling the Soho Theatre for a run of five nights. The show was a selection of their favourite bits of the last two shows toured, with each of the three taking their turn to do their thing.</p>
<p>The first thing to strike us was the similarity of the crowd, both in type and behaviour, to the crowds attending any of the other comedy gigs here. Nerdery is no longer a niche market — we're all nerds now. Once ushered into the theatre we were presented with a
slick show, with (slightly irritating) sound effects and power point slides, and a number of health-and-safety-bending experiments. A fantastically well-rehearsed team worked together to facilitate each others' performances.</p>
<div style="float:right; margin-left: 1em;">
<iframe width="400" height="225" src="https://www.youtube.com/embed/_dQJBBklpQQ" frameborder="0" allowfullscreen></iframe></div>
<p>Mould started with his speciality — the <em>Mould effect</em> — where a chain of beads pours out of a jar, seemingly defying gravity by rising higher and higher in the air before clattering back down to the stage. With effortless humour Mould explained the journey from discovering this phenomenon (in possibly dubious circumstances), to becoming a YouTube sensation, finding a scientific explanation and finally immortality in the effect being named after him. A final demonstration showed that there is always room for dub step to dial up the appreciation of any scientific phenomenon to 11. </p>
<p>Parker represented the mathsy part of the show, relentlessly exploiting mathematics' role as the queen of nerdery, but not without surprises. Parker proved we are all obsessed with spreadsheets, even if we don't realise it, by brilliantly showing our addiction to all things screens is really just an ongoing worship of Excel spreadsheets. And we particularly liked his great demonstration of parabolic mirrors as he sent heat across the stage causing a remote fire —
all in honour of one of our own favourite mathematical shapes, the parabola. </p>
<p>Arney's electric ukulele and heavenly voice explored love scientifically and mathematically with a suitable helping of dark humour. The final number showed how well these three performers work together: Arney playing a maths based love song, supported by competing visual gags from Parker on an OHP and Mould on Power Point slides. It's always impressive when something is that slick but so relaxed that you can't tell which bits are improvised and which are rehearsed. Definitely worth seeing!</p></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">Review by </div><div class="field-items"><div class="field-item even">Rachel Thomas with Marianne Freiberger</div></div></div>Tue, 17 Jan 2017 13:31:22 +0000mf3446770 at https://plus.maths.org/contenthttps://plus.maths.org/content/node/6770#comments