In the latest poll of our Science fiction, science fact project you told us that you wanted to know how many dimensions there are. Here is the first part of theoretical physicist David Berman's answer (you can read the second part here). Click here to see other articles exploring this question and here to listen to our interview with Berman as a podcast.
What is a dimension?
What do we mean by "dimension"? An intuitive description of dimensions is that it's the ways in which we can move – up and down, left and right, forward and backwards – which leads us to believe we live in a three-dimensional world.
35th Street and 10th Avenue – a two dimensions address in New York.
Let's think about what this means in terms of giving someone an address. You can specify where you are in two dimensions. For example in New York you'd say "on 35th Street and 10th Avenue", a simple two dimensional address of how many blocks north-south and how many blocks east-west you are.
But what if you are on a particular floor of a tall office block? Then you'd have to say "5th floor, on 35th Street and 10th Avenue". You have specified three things – how far north or south, how far east or west, and how far up – so you have given three coordinates, specifying a point in three-dimensional space.
In fact we are used to living in four dimensions. If you are arranging a meeting then you need to give one more piece of information: the time of the meeting, say 3pm. With four coordinates you can describe any event. We don't tend to clump time with the spatial dimensions but if you think about it, any event really happens in a spacetime with four dimensions. You can measure differences in time just like you can measure differences in space – we measure differences in space with a ruler and we measure differences in time with a clock. So anything you can think of in terms of space you can think of in terms of time.
We can learn a lesson from this: our human perception of dimensions is limited. We only perceive three dimensions. We can understand that time is an extra dimension, an extra location for any event, but we don't see time in the same way. We experience it very differently from spatial dimensions as human beings. This hints that human perceptions of things may not be the end of the story in terms of what's possible.
A point in 16-dimensional space
Mathematically the number of dimensions is just the number of coordinates you need to specify a point. We're familiar with specifying points in four-dimensional spacetime, but can you imagine a space where you need five bits of information? Or six? Or seven…? Mathematicians regularly work in higher dimensional spaces, but they are not the only ones. For example if you're a sound mixer making music you might be working with 12, 24, or even 128 tracks. At each moment in the music each of the, say, 24 tracks has a specific volume, defining a point in a 24-dimensional space of sound. The number of dimensions is just the number of bits of information you need to specify a point.
Curved and hidden
Einstein famously combined the three spatial dimensions with additional dimension of time into four-dimensional spacetime. But at around the same time the German mathematician Theodor Kaluza and the Swedish physicist Oskar Klein began to wonder if there were more than three spatial dimensions.
General relativity, which was Einstein's big idea, said that dimensions could be curved. This sounds strange at first but we are used to that as well. The surface of the Earth is two dimensional and it's curved. We specify a location on the surface of the Earth using two coordinates: longitude and latitude. But these coordinates are periodic – you can come back to yourself if you travel 360 degrees around the Earth. The surface of the Earth is a two-dimensional surface inside a three-dimensional space.
Kaluza and Klein wondered if physical space could be curved in a similar way. That particular dimensions don't go on forever in space but the dimensions of space itself could be so curved that space comes back on itself like a circle.
We're already used to curved dimensions – the two dimensional surface of the Earth is curved with periodic coordinates.(Image from NASA)
That sounds a bit weird: could you go off in your rocket ship in a particular direction and then arrive back where you started, just like circling the earth in a plane? Would we notice these curved dimensions?
This would depend on the size of the circle around which the dimension curves. If the circle was, say a hundredth or a thousandth the size of an atom, then that is so small you probably wouldn't even notice the presence of that dimension. Imagine rolling up a piece of paper very tightly into a cylinder with a tiny diameter – from a distance you wouldn't notice that it was a cylinder at all, it would look just like a line.
Thinking in terms of coordinates, this would be just like when giving an address in three-dimensional space using only two pieces of information. If my house is located at, say, 4th and 5th streets, and is one step up from the road, I don't normally specify that because it won't really be noticed. If you're travelling four streets one way and five streets another and you come to the house and it's 10 centimetres above the ground because of some tiny step, it's irrelevant to the big picture.
Kaluza and Klein proposed that you could have one extra spatial dimension but if that dimension was curled up into a circle with a very small radius, smaller than any other scale that was around, then you wouldn't notice it.
Gravity in the fourth dimension
So why should we care about an extra dimension if you can't even notice it? Kaluza and Klein showed, using general relativity, that this extra dimension would still have an effect on the space around us. In particular they showed that the effect of gravity in that very small fifth dimension would actually appear to us, from our larger-scale perspective, as electromagnetism.
This Kaluza-Klein Theory had many fascinating consequences. Things that have charge under electromagnetism would just be things that are moving in this circular fifth dimension. The radius of this extra dimension would be related to the electric charge of particles. A charged particle would move in one direction in the fifth dimension, and an oppositely charged particle would move in the other. And as the momentum of two oppositely charge particles would cancel out when they collide, they would no longer move in the fifth dimension making their combined charge neutral.
Although Kaluza and Klein's theory seemed to unify the forces of electromagnetism and gravity it wasn't widely accepted when it was published in 1921. It was mathematically true but the concept of extra spatial dimensions seemed very strange. However some 60 years later this mathematical oddity provided a missing piece in the hunt for quantum gravity…
Read more in the next article: The ten dimensions of string theory.
About the author
David Berman is a Reader in Theoretical Physics at Queen Mary, University of London. He previously spent time at the universities of Manchester, Brussels, Durham, Utrecht, Groningen, Jerusalem and Cambridge as well as a year at CERN in Geneva.
His interests outside of physics include football, music and theatre and the arts.
Thankyou for this article. Very interesting and clearly written. May I suggest a proofread as there are several mistakes which distract from the polish and may confuse a reader for whom English is not their native language.
Thanks for pointing this out, we will check the article.
in sentence below, there are extra verbs to remove, due to proper english."Things that have charge under electromagnetism (are) would just be things that are moving in this circular fifth dimension. The radius of this extra dimension would be related to the electric charge of particles. A charged particle would (be) move in one direction in the fifth dimension" removing the two words verbs are and be as marked/
Excellent article. I did not notice any mistakes at all...
That's because you are in the future!
Katika utafiti wa hivi karibuni wa Science Fiction, Science Fact Project, tulijifunza kwamba kuna vipimo vingi vya ulimwengu. Hapa ni sehemu ya kwanza ya jibu la mwanafizikia wa nadharia David Berman (unaweza kusoma sehemu ya pili hapa). Bonyeza hapa ili kuona makala nyingine zinazochunguza swali hili na hapa kusikiliza mahojiano yetu na Berman kama podcast. Mwelekeo ni nini? Tunamaanisha nini kwa "upana"? "Ufafanuzi wa intuitive wa vipimo ni kwamba ni njia ambazo tunaweza kusonga ""juu na chini, kushoto na kulia, mbele na nyuma"" ambayo inatuongoza kuamini kwamba tunaishi katika ulimwengu wa vipimo vitatu." 35th Street na 10th Avenue - New York Hebu fikiria nini hii ina maana katika suala la kutoa mtu anwani. Unaweza kutaja mahali ulipo katika vipimo viwili. "Kwa mfano, katika New York, unaweza kusema ""katika 35th Street na 10th Avenue,"" anwani rahisi ya pande mbili ya jinsi vitalu kaskazini-kusini na jinsi vitalu mashariki-magharibi wewe ni." Lakini vipi ikiwa uko kwenye orofa fulani ya jengo refu la ofisi? "Kama unavyoona, ""Sasa, katika nyumba ya 5th, katika barabara ya 35th na 10th Avenue.""" Umefafanua mambo matatu - ni mbali kadiri gani kaskazini au kusini, ni mbali kadiri gani mashariki au magharibi, na ni mbali kadiri gani juu - kwa hiyo umetoa kuratibu tatu, ukitaja uhakika katika nafasi ya pande tatu. Kwa kweli tumezoea kuishi katika vipimo vinne. Ikiwa una mpango wa mkutano, unapaswa kutoa taarifa ya muda wa mkutano, kwa mfano, saa 3 jioni. Kwa kuratibu nne unaweza kuelezea tukio lolote. Wakati haujumuishi vipimo vya anga, lakini ikiwa utafikiria juu yake, tukio lolote linafanyika katika nafasi-wakati na vipimo vinne. Unaweza kupima tofauti katika muda kama vile unaweza kupima tofauti katika nafasi <unk> sisi kupima tofauti katika nafasi na mtawala na sisi kupima tofauti katika muda na saa. Kwa hivyo chochote unachoweza kufikiria katika suala la nafasi unaweza kufikiria katika suala la wakati. Tunaweza kujifunza somo kutokana na hili: utambuzi wetu wa kibinadamu wa vipimo ni mdogo. Tunaona vipimo vitatu tu. Wakati ni kipimo cha ziada, eneo la ziada kwa tukio lolote, lakini hatuoni wakati kwa njia ileile. Sisi uzoefu ni tofauti sana kutoka vipimo nafasi kama wanadamu. Hii inaonyesha kwamba maoni ya binadamu ya mambo yanaweza kuwa si mwisho wa hadithi katika suala la nini inawezekana. Katika nafasi ya 16 ya mwelekeo, idadi ya vipimo ni idadi ya kuratibu zinazohitajika kuelezea sehemu. Kwa mfano, unaweza kuamua nafasi ya nafasi ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 4 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 ya 5 Au ni sita? Au saba...? Wanahisabati hufanya kazi kwa ukawaida katika nafasi zenye vipimo vya juu zaidi, lakini si wao pekee. Kwa mfano, kama wewe ni mchanganyiko wa sauti, unaweza kufanya kazi na nyimbo 12, 24 au hata 128. Kwa mfano, katika muziki wa muziki wa muziki, kila moja ya nyimbo 24 zina sauti maalum, inayofafanua hatua katika nafasi ya sauti ya 24-dimensional. Idadi ya vipimo ni tu idadi ya bits ya habari unahitaji kutaja uhakika. "Ashtine aliunganisha ""mipangilio mitatu ya nafasi"" na ""mipangilio ya ziada ya wakati"" katika nafasi-wakati ya vipimo vinne." Katika miaka ya mwisho ya karne ya 20, mwanahisabati Mjerumani Theodor Kaluza na mwanafizikia Mhispania Oskar Klein walianza kugundua kwamba kuna mambo mengi zaidi ya vipimo vitatu vya anga. Nadharia ya jumla ya upendeleo, ambayo ilikuwa wazo kubwa la Einstein, ilisema kwamba vipimo vingeweza kupindika. Hii inasikika ya ajabu kwa mara ya kwanza lakini sisi ni kutumika kwa kwamba pia. Uso wa dunia ni wa pande mbili na ni mkondo. Tunataja eneo kwenye uso wa Dunia kwa kutumia kuratibu mbili: longitudo na latitudo. "Kama unavyoona, ""mipangilio"" hii ni ya kawaida, na unaweza kurudi kwa wewe mwenyewe ikiwa unasafiri digrii 360 kuzunguka Dunia." Uso wa Dunia ni uso wa pande mbili ndani ya nafasi ya pande tatu. Kaluza na Klein walijiuliza ikiwa nafasi ya kimwili ingeweza kupindika kwa njia kama hiyo. """Urefu wa kipekee haupiti milele katika nafasi, lakini urefu wa nafasi yenyewe unaweza kupindika sana hivi kwamba nafasi inarudi juu yake mwenyewe kama mduara.""" Sisi tayari kutumika kwa vipimo curved <unk> uso mbili-dimensional ya Dunia ni curved na kuratibu mara kwa mara. "Kuna kitu kinachotokea wakati wa ""Misa"" ya NASA: unaweza kwenda kwa ndege yako ya roketi kwa mwelekeo fulani na kisha kurudi mahali ulipoanza, kama vile kuzunguka dunia kwa ndege?" Je, tungetambua vipimo hivi vilivyojipinda? Hii itategemea ukubwa wa mduara ambao mwinuko wa kipimo. Kwa mfano, kama mduara ni ndogo kama sehemu ya mia moja au elfu moja ya atomu, basi ni ndogo sana hivi kwamba mtu hawezi hata kugundua kuwapo kwa sehemu hiyo. Fikiria rolling juu kipande cha karatasi tightly sana katika silinda na kipenyo kidogo <unk> kutoka umbali huwezi kutambua kwamba ilikuwa silinda wakati wote, ingekuwa kuangalia tu kama mstari. Kwa kuzingatia miktadha, hii itakuwa kama kutoa anwani katika nafasi ya pande tatu kwa kutumia vipande viwili tu vya habari. Ikiwa nyumba yangu iko, tuseme, kwenye barabara ya nne na ya tano, na ni hatua moja juu kutoka barabara, kwa kawaida sielewi kwamba kwa sababu haitaonekana. Kama wewe ni kusafiri mitaano nne moja njia na mitaano tano nyingine na wewe kuja nyumba na ni sentimeta kumi juu ya ardhi kwa sababu ya hatua ndogo, ni muhimu kwa picha kubwa. Kaluza na Klein walipendekeza kwamba unaweza kuwa na mwelekeo mmoja wa ziada wa anga, lakini ikiwa mwelekeo huo ungepigwa na mviringo wenye mduara mdogo sana, mdogo kuliko kiwango kingine chochote kilichokuwa karibu, basi usingeona. Gravity in the fourth dimension: Kwa nini tuwe na wasiwasi kuhusu dimension ya ziada ikiwa huwezi hata kuiona? Kaluza na Klein walionyesha, kwa kutumia nadharia ya jumla ya upendeleo, kwamba mwelekeo huu wa ziada bado ungekuwa na athari kwenye nafasi inayotuzunguka. Kwa mfano, kuna uwezekano kwamba nguvu ya uvutano katika mwelekeo wa tano wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu wa ulimwengu. Nadharia hii ya Kaluza-Klein ilikuwa na matokeo mengi yenye kuvutia. Vitu ambavyo vina malipo chini ya sumaku ya umeme vitakuwa tu vitu ambavyo vinaenda katika mwendo huu wa mviringo wa tano. Kiwango cha kipimo hiki cha ziada kingehusiana na mzigo wa umeme wa chembechembe. Chembe yenye nishati ingeenda katika mwelekeo mmoja katika mwelekeo wa tano, na chembe yenye nishati kinyume ingeenda katika mwelekeo mwingine. Na kama kasi ya chembe mbili zenye malipo tofauti ingekamilika wakati zinapogongana, hazingehama tena katika mwelekeo wa tano, na kufanya malipo yao ya pamoja kuwa ya moja kwa moja. Ingawa nadharia ya Kaluza na Klein ilionekana kuunganisha nguvu za umeme na mvuto haikukubaliwa sana wakati ilichapishwa mnamo 1921. Ilikuwa kweli kisayansi lakini dhana ya vipimo vya ziada vya nafasi ilionekana kuwa ya ajabu sana. Hata hivyo, miaka 60 baadaye, jambo hilo la ajabu la hisabati lilitusaidia kupata chanzo cha nguvu ya uvutano wa quantum. Kuhusu mwandishi David Berman ni Msomaji katika Theoretical Physics katika Malkia Mary, Chuo Kikuu cha London. Alifanya kazi katika vyuo vikuu vya Manchester, Brussels, Durham, Utrecht, Groningen, Jerusalem na Cambridge na mwaka mmoja katika CERN huko Geneva. Maslahi yake nje ya fizikia ni pamoja na mpira wa miguu, muziki na ukumbi wa michezo na sanaa. Asanteni kwa makala hii. Ni jambo lenye kupendeza sana na limeandikwa kwa uwazi. Naomba uisome kwa makini kwa kuwa kuna makosa kadhaa ambayo yanakengeusha fikira kutoka kwa lugha ya Kipolishi na yanaweza kumchanganya msomaji ambaye Kiingereza si lugha yake ya asili. Asanteni kwa kuashiria hili, tutaangalia makala hiyo. Katika sentensi hapa chini, kuna vitenzi vya ziada vya kuondoa, kwa sababu ya Kiingereza sahihi. "Vitu ambavyo vina malipo chini ya sumaku ya umeme (ni) vitakuwa tu vitu ambavyo vinaenda katika mwendo huu wa mviringo wa tano. Kiwango cha kipimo hiki cha ziada kingehusiana na mzigo wa umeme wa chembechembe. "Kama vile chembe ya umeme inakwenda katika mwelekeo mmoja katika mwelekeo wa tano, maneno ""ni"" na ""kuwa"" yanaondolewa kama ilivyoonyeshwa." Sijaona makosa yoyote... hiyo ni kwa sababu wewe ni katika siku zijazo!