{"id":451,"date":"2019-02-03T01:04:48","date_gmt":"2019-02-03T00:04:48","guid":{"rendered":"https:\/\/www.leotheodon.com\/V4\/?p=451"},"modified":"2019-02-03T01:10:26","modified_gmt":"2019-02-03T00:10:26","slug":"volume-de-lhypersphere","status":"publish","type":"post","link":"https:\/\/www.leotheodon.com\/V4\/2019\/02\/03\/volume-de-lhypersphere\/","title":{"rendered":"Volume de l’hypersph\u00e8re"},"content":{"rendered":"\n

<\/p>\n\n\n\n

Une hypersph\u00e8re est une sph\u00e8re en \"D\" dimensions. Elle g\u00e9n\u00e9ralise le concept de sph\u00e8re que nous connaissons tous en dimension 3. L’id\u00e9e est ici de donner une expression pour le volume \"V_D\" d’une hypersph\u00e8re de rayon \"R\" dans un espace \u00e0 \"D\" dimensions.<\/p>\n\n\n\n

Tout d’abord, on peut \u00e9crire que \"V_D = C_D \times R^D\", o\u00f9 \"C_D\" est le volume de l’hypersph\u00e8re unit\u00e9 (hypersph\u00e8re de rayon \"R=1\").<\/p>\n\n\n\n

Remarque :<\/strong>
On en d\u00e9duit imm\u00e9diatement l’expression de \"S_D\", la surface de l’hypersph\u00e8re de rayon \"R\". En d\u00e9rivant l’expression pr\u00e9c\u00e9dente par rapport \u00e0 \"R\", on obtient: \"S_D = D.C_D.R^{D-1}\".
<\/p>\n\n\n\n

<\/p>\n\n\n\n

Calculons \u00e0 pr\u00e9sent \"C_D\". Pour ce faire, nous allons poser \"I_D = \displaystyle \prod_{i=1}^D \left( \int_{-\infty}^{+\infty} e^{-x_{i}^2} \right)\". Tout d’abord, on peut remarquer que l’on a affaire \u00e0 un produit d’int\u00e9grales de Gaussiennes. On a donc :<\/p>\n\n\n\n

<\/a>

(1) <\/span>   <\/span>\"\begin{equation*}<\/p><\/p>\n\n\n\n

Nous allons \u00e0 pr\u00e9sent effectuer le changement de variables \"\displaystyle \sum_{i=1}^D x_{i}^2 = R^2\" et \"\displaystyle \prod_{i=1}^D dx_i = dV_D\". On peut d\u00e8s lors \u00e9crire la relation suivante :<\/p>\n\n\n\n

  <\/span>   <\/span>\"\begin{align*} I_D = \displaystyle \prod_{i=1}^D \left( \int_{-\infty}^{+\infty} e^{-x_{i}^2} \right) & = \int_{-\infty}^{+\infty} e^{-(x_{1}^2+x_{2}^2+…+x_{D}^2)} dx_1dx_2…dx_D\\ & = \int_{0}^{+\infty} e^{-R^2}dV_D \end{align*}\"<\/p><\/p>\n\n\n\n

On peut alors effectuer le nouveau changement de variable \"y=R^2\". En particulier, on peut remarquer que l’on a la relation suivante :<\/p>\n\n\n\n

<\/a>

(2) <\/span>   <\/span>\"\begin{equation*}<\/p><\/p>\n\n\n\n

On en d\u00e9duit alors imm\u00e9diatement la nouvelle expression de \"I_D\", c’est-\u00e0-dire :<\/p>\n\n\n\n

<\/a>

(3) <\/span>   <\/span>\"\begin{align*} I_D & = \displaystyle \dfrac{D}{2}C_D \int_{0}^{+\infty} y^{\frac{D}{2}-1}e^{-y}dy\\& = \dfrac{D}{2}C_D \Gamma\left(\frac{D}{2}\right)\\& = \dfrac{D}{2}C_D \left(\frac{D}{2}-1\right)!\\& = C_D \left(\frac{D}{2}\right)! \end{align*}\"<\/p><\/p>\n\n\n\n

Ainsi, on obtient le volume de la sph\u00e8re unit\u00e9 :<\/p>\n\n\n\n

<\/a>

(4) <\/span>   <\/span>\"\begin{equation*}<\/p><\/p>\n\n\n\n

Ainsi, de mani\u00e8re g\u00e9n\u00e9rale, on a l’expression suivante pour le volume de l’hypersph\u00e8re en dimension \"D\" :<\/p>\n\n\n\n

(5) <\/span>   <\/span>\"\begin{equation*}V_D = \dfrac{\sqrt{\pi}^D}{\left(\frac{D}{2}\right)!}R^D\end{equation*}\"<\/p><\/p>\n\n\n\n

Remarque :<\/strong>
On remarquer que si l’on calcule le volume de l’hypersph\u00e8re unit\u00e9, ce dernier atteint un maximum pour \"D=5\". De plus, il est int\u00e9ressant de voir que le volume de l’hypersph\u00e8re tend vers 0 quand le nombre de dimension tend vers l’infini quand le volume de l’hypercube, quant \u00e0 lui, \u00e9gal \u00e0 \"2^D\" tend vers l’infini.<\/p>\n\n\n\n
Dimension<\/strong><\/td>1<\/strong><\/td>2<\/strong><\/td>3<\/strong><\/td>4<\/strong><\/td>5<\/strong><\/td>6<\/strong><\/td><\/tr>
Sph\u00e8re unit\u00e9<\/strong><\/td>\"2\"<\/td>\"\pi\"<\/td>\"\dfrac{4\pi}{3}\"<\/td>\"\dfrac{\pi^2}{4}\" <\/td>\"\dfrac{8\pi^2}{15}\"
<\/td>		\"\dfrac{\pi^3}{6}\" <\/td><\/tr><\/tbody><\/table>\n\n\n\n
T\u00e9l\u00e9charger au format pdf :<\/a>T\u00e9l\u00e9charger<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"

Une hypersph\u00e8re est une sph\u00e8re en D dimensions. Elle g\u00e9n\u00e9ralise le concept de sph\u00e8re que nous connaissons tous en dimension 3. L\u2019id\u00e9e est ici de donner une expression pour le volume d\u2019une hypersph\u00e8re de rayon…<\/p>\n","protected":false},"author":1,"featured_media":329,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"rop_custom_images_group":[],"rop_custom_messages_group":[],"rop_publish_now":"initial","rop_publish_now_accounts":[],"rop_publish_now_history":[],"rop_publish_now_status":"pending","_exactmetrics_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0,"footnotes":""},"categories":[3,5],"tags":[56,54,16,55],"class_list":["post-451","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-education","category-math","tag-geometrie","tag-hypersphere","tag-math","tag-sphere"],"_links":{"self":[{"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/posts\/451","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/comments?post=451"}],"version-history":[{"count":23,"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/posts\/451\/revisions"}],"predecessor-version":[{"id":475,"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/posts\/451\/revisions\/475"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/media\/329"}],"wp:attachment":[{"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/media?parent=451"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/categories?post=451"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/tags?post=451"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}