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四维超立方体与三维空间的交集演示

四维超立方体与三维空间的交集演示
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2020-06-10 18:07
向量立体几何几何作图
①本范例演示了三维空间在穿过四维超立方体过程中的交集形态；②涉及核心数学概念包括超立方体、维度、交集、几何可视化；③作图要点提炼自构造摘要，使用了Ellipse（椭圆）、Point（点）、Angle（角度）、If（条件）、Join（连接）、Segment（线段）、Vector（向量）、Intersect（交集）、Circle（圆）、Line（直线）、Conic（圆锥曲线）、Dilate（缩放）、Max（最大值）等GeoGebra命令/对象；④适合高中及以上数学学习者，用于理解高维几何与交集概念。

GeoGebra 命令及参数

  1. Ellipse(O_r - (0, 0.95), O_r + (0, 0.95), 1) → mira2
  2. Point(mira2) → B
  3. Angle(M_3, O_r, B) → θ
  4. Ellipse(O_r - (0.95, 0), O_r + (0.95, 0), 1) → mira1
  5. Point(mira1) → A
  6. Angle(U_1, O_r, A) → φ
  7. If(t > 2, t - 2, t) → tt
  8. If(tt < 1, tt, 1) → tt1
  9. If(tt ≥ 1, tt - 1, 0) → tt2
  10. Angle(U_1, O_r, AM) → φ2
  11. Angle(M_3, O_r, BM) → θ2
  12. Join({{x0, y0, z0}} * MM') → M
  13. If(iguales, 1, 0.5 + cos(φ)^(2)) → hh
  14. If(iguales, hh, 2 - hh) → gg
  15. Join({{1, 1, -1} * hh} * MM) → MV1
  16. Join({{-1, -1, -1} * hh} * MM) → MV8
  17. Join({{-1, -1, 1} * hh} * MM) → MV7
  18. Join({{-1, 1, -1} * hh} * MM) → MV6
  19. Join({{-1, 1, 1} * hh} * MM) → MV5
  20. Join({{1, -1, -1} * hh} * MM) → MV4
  21. Join({{1, 1, 1} * hh} * MM) → MV2
  22. Join({{1, -1, 1} * hh} * MM) → MV3
  23. Segment(CV7, CV6) → r_{13}
  24. Vector(O_r, O) → u
  25. Intersect(mira1, mira2) → Mira1, Mira2, Mira3, Mira4
  26. Circle(O, Scale) → c
  27. Line(O, yAxis) → b
  28. Segment(Mira3, Mira2) → k_{13}
  29. Segment((-1, 0), U_1) → l
  30. Segment((0, -1), M_3) → m
  31. Angle(V3, V2, V4) → cub1Ang
  32. Angle(V6, V2, V5) → cub3Ang
  33. Angle(V5, V7, V6) → cub5Ang
  34. Angle(V4, V7, V3) → cub6Ang
  35. Angle(V5, V2, V7) → cub2Ang
  36. Angle(V4, V1, V8) → cub4Ang
  37. Segment(V1, V2) → cubo1A1
  38. Segment(V2, V3) → cubo1A2
  39. Segment(V3, V4) → cubo1A3
  40. Segment(V4, V1) → cubo1A4
  41. Segment(V1, V6) → cubo1A5
  42. Segment(V6, V8) → cubo1A6
  43. Segment(V8, V4) → cubo1A7
  44. Segment(V2, V5) → cubo1A8
  45. Segment(V5, V7) → cubo1A9
  46. Segment(V7, V3) → cubo1A10
  47. Segment(V6, V5) → cubo1A11
  48. Segment(V7, V8) → cubo1A12
  49. Segment(P, CV3) → rayo3
  50. Segment(P, CV2) → rayo2
  51. Segment(P, CV5) → rayo5
  52. Segment(P, CV7) → rayo7
  53. Segment(P, CV4) → rayo4
  54. Segment(P, CV8) → rayo8
  55. Segment(P, CV6) → rayo6
  56. Segment(P, CV1) → rayo1
  57. Segment(CV3, CV5) → f
  58. Segment(CV7, CV2) → h1
  59. Segment(CV3, CV8) → d_{13}
  60. Segment(CV7, CV4) → i_{13}
  61. Segment(CV2, CV6) → n
  62. Segment(CV5, CV1) → p
  63. Segment(CV8, CV5) → q_2
  64. Segment(CV3, CV1) → k_1
  65. Segment(CV4, CV2) → l_1
  66. Segment(CV8, CV1) → m_1
  67. Segment(CV6, CV4) → n_1
  68. Segment(E11, E12) → a_9
  69. Segment(E12, E14) → b_9
  70. Segment(E14, E13) → c_9
  71. Segment(E11, E13) → d_9
  72. Segment(E15, E17) → e_9
  73. Segment(E17, E18) → f_9
  74. Segment(E16, E18) → g_9
  75. Segment(E16, E15) → h_9
  76. Segment(E11, E15) → i_9
  77. Segment(E12, E16) → j_9
  78. Segment(E18, E14) → k_9
  79. Segment(E17, E13) → l_9
  80. Conic(C1, C2, C3, C4, C5) → luz
  81. Line(O, j) → a
  82. Line(O, i) → g
  83. Line(O, k) → j_{13}
  84. Dilate(luz, 0.05, P) → luz1
  85. Dilate(luz, 0.045, P) → luz2
  86. Dilate(luz, 0.04, P) → luz3
  87. Dilate(luz, 0.035, P) → luz4
  88. Dilate(luz, 0.03, P) → luz5
  89. Dilate(luz, 0.025, P) → luz6
  90. Dilate(luz, 0.02, P) → luz7
  91. Dilate(luz, 0.015, P) → luz8
  92. Dilate(luz, 0.01, P) → luz9
  93. Dilate(luz, 0.005, P) → luz10
  94. Segment(CV7, CV3) → linea29
  95. Segment(CV7, CV5) → linea05
  96. Segment(CV5, CV2) → linea14
  97. Segment(CV3, CV2) → linea16
  98. Segment(CV7, CV8) → linea31
  99. Segment(CV3, CV4) → linea0304
  100. Segment(CV5, CV6) → linea0506
  101. Segment(CV2, CV1) → linea08
  102. Segment(CV8, CV6) → linea10
  103. Segment(CV6, CV1) → linea12
  104. Segment(CV1, CV4) → linea18
  105. Segment(CV8, CV4) → linea19
  106. Segment(B_2, C_2) → n_{14}
  107. Segment(D_2, E_2) → p_{14}
  108. Segment(F_2, G_2) → q_{14}
  109. Segment(H_2, I_2) → r_{14}
  110. Segment(C_2, D_2) → s_{14}
  111. Segment(E_2, F_2) → t_{14}
  112. Segment(G_2, H_2) → a_{15}
  113. Segment(I_2, J_2) → b_{15}
  114. Segment(V9, V10) → cubo2A13
  115. Segment(V15, V11) → cubo2A22
  116. Segment(V14, V13) → cubo2A23
  117. Segment(V15, V16) → cubo2A24
  118. Segment(V10, V11) → cubo2A14
  119. Segment(V11, V12) → cubo2A15
  120. Segment(V12, V9) → cubo2A16
  121. Segment(V9, V14) → cubo2A17
  122. Segment(V14, V16) → cubo2A18
  123. Segment(V16, V12) → cubo2A19
  124. Segment(V10, V13) → cubo2A20
  125. Segment(V13, V15) → cubo2A21
  126. Segment(V1, V9) → cubo3A25
  127. Segment(V2, V10) → cubo3A26
  128. Segment(V3, V11) → cubo3A27
  129. Segment(V4, V12) → cubo3A28
  130. Segment(V6, V14) → cubo4A30
  131. Segment(V5, V13) → cubo4A29
  132. Segment(V7, V15) → cubo4A31
  133. Segment(V8, V16) → cubo4A32
  134. Join({{1 + h3i, 1 + h3j, 1 + h3k} * gg} * MM) → MV10
  135. Join({{1 + h3i, 1 + h3j, -1 + h3k} * gg} * MM) → MV9
  136. Join({{1 + h3i, -1 + h3j, 1 + h3k} * gg} * MM) → MV11
  137. Join({{1 + h3i, -1 + h3j, -1 + h3k} * gg} * MM) → MV12
  138. Join({{-1 + h3i, 1 + h3j, 1 + h3k} * gg} * MM) → MV13
  139. Join({{-1 + h3i, 1 + h3j, -1 + h3k} * gg} * MM) → MV14
  140. Join({{-1 + h3i, -1 + h3j, 1 + h3k} * gg} * MM) → MV15
  141. Join({{-1 + h3i, -1 + h3j, -1 + h3k} * gg} * MM) → MV16
  142. Segment(P, CV9) → rayo9
  143. Segment(P, CV12) → rayo12
  144. Segment(P, CV14) → rayo14
  145. Segment(P, CV15) → rayo15
  146. Segment(P, CV11) → rayo11
  147. Segment(P, CV13) → rayo13
  148. Segment(P, CV10) → rayo10
  149. Segment(P, CV16) → rayo16
  150. Segment(CV15, CV13) → linea06
  151. Segment(CV13, CV14) → linea09
  152. Segment(CV16, CV15) → linea11
  153. Segment(CV16, CV14) → linea13
  154. Segment(CV15, CV7) → linea15
  155. Segment(CV13, CV5) → linea17
  156. Segment(CV6, CV14) → linea20
  157. Segment(CV8, CV16) → linea21
  158. Segment(CV12, CV9) → linea22
  159. Segment(CV9, CV10) → linea23
  160. Segment(CV11, CV10) → linea24
  161. Segment(CV11, CV12) → linea25
  162. Segment(CV10, CV2) → linea26
  163. Segment(CV11, CV3) → linea27
  164. Segment(CV4, CV12) → linea28
  165. Segment(CV1, CV9) → linea30
  166. Segment(CV14, CV9) → linea32
  167. Segment(CV10, CV13) → linea1013
  168. Segment(CV11, CV15) → linea04
  169. Segment(CV16, CV12) → linea07
  170. If(t < 1, t, 1) → t1
  171. If((t ≥ 1) ∧ (t ≤ 2), t - 1, If[t < 1, 0, 1]) → t2
  172. Segment(N1103, N1110) → n1
  173. Segment(N1110, N1112) → n2
  174. Segment(N1103, N1112) → n3
  175. Segment(N1103, N1115) → n4
  176. Segment(N1115, N1112) → n5
  177. Segment(N1115, N1110) → n6
  178. If((t ≥ 2) ∧ (t ≤ 3), t - 2, If[t < 2, 0, 1]) → t3
  179. If((t ≥ 3) ∧ (t ≤ 4), t - 3, If[t < 3, 0, 1]) → t4
  180. If((t ≥ 4) ∧ (t ≤ 5), t - 4, If[t < 4, 0, 1]) → t5
  181. Segment(N1516, N1513) → n7
  182. Segment(N1513, N1507) → n8
  183. Segment(N1507, N1516) → n9
  184. Segment(N0302, N0304) → n10
  185. Segment(N0304, N0307) → n11
  186. Segment(N0307, N0302) → n12
  187. Segment(N1013, N1002) → n13
  188. Segment(N1013, N1009) → n14
  189. Segment(N1002, N1009) → n15
  190. Segment(N1216, N1209) → n16
  191. Segment(N1204, N1216) → n17
  192. Segment(N1204, N1209) → n18
  193. Segment(N1209, N1009) → n22
  194. Segment(N1513, N1013) → n23
  195. Segment(N1507, N0307) → n24
  196. Segment(N1516, N1216) → n19
  197. Segment(N0302, N1002) → n21
  198. Angle(N1110, N1112, N1103) → angPoli1
  199. Angle(N1110, N1103, N1115) → angPoli4
  200. Angle(N1115, N1112, N1110) → angPoli3
  201. Angle(N1103, N1112, N1115) → angPoli2
  202. Angle(N1216, N1209, N1513) → angPoli11
  203. Angle(N1013, N1009, N1002) → angPoli6
  204. Angle(N1013, N1002, N1507) → angPoli9
  205. Angle(N1204, N1209, N1216) → angPoli7
  206. Angle(N1204, N1216, N1507) → angPoli12
  207. Angle(N1209, N1204, N0302) → angPoli10
  208. Angle(N0302, N0304, N0307) → angPoli5
  209. Angle(N1513, N1507, N1516) → angPoli8
  210. Segment(V6, V11) → diagonalPrincipal
  211. Vector((N1103 + N1110 + N1112) / 3, (N1103 + N1110 + N1112 + Element[tetraL1, 1] * i + Element[tetraL1, 2] * j + Element[tetraL1, 3] * k) / 3) → zzVectorTetra1
  212. Vector((N1103 + N1115 + N1112) / 3, (N1103 + N1115 + N1112 + Element[tetraL2, 2] * i + Element[tetraL2, 2] * j + Element[tetraL2, 3] * k) / 3) → zzVectorTetra2
  213. Vector((N1112 + N1110 + N1115) / 3, (N1112 + N1110 + N1115 + Element[tetraL3, 1] * i + Element[tetraL3, 2] * j + Element[tetraL3, 3] * k) / 3) → zzVectorTetra3
  214. Segment(N0901, N0914) → n42
  215. Segment(N1608, N1614) → n27
  216. Segment(N1305, N1314) → n33
  217. Segment(N0705, N0708) → n34
  218. Segment(N0205, N0201) → n36
  219. Segment(N0408, N0401) → n38
  220. Segment(N0205, N1305) → n32
  221. Segment(N0705, N1305) → n29
  222. Segment(N0705, N0205) → n31
  223. Segment(N1314, N1614) → n30
  224. Segment(N1314, N0914) → n37
  225. Segment(N1614, N0914) → n28
  226. Segment(N0201, N0401) → n39
  227. Segment(N0401, N0901) → n41
  228. Segment(N0201, N0901) → n40
  229. Segment(N0408, N1608) → n26
  230. Segment(N0708, N1608) → n25
  231. Segment(N0708, N0408) → n35
  232. Segment(N1406, N0506) → n45
  233. Segment(N0506, N0106) → n44
  234. Segment(N0106, N1406) → n48
  235. Segment(N0506, N0806) → n43
  236. Segment(N0806, N0106) → n47
  237. Segment(N0806, N1406) → n46
  238. Angle(N0708, N0408, N1608) → angPoli16
  239. Angle(N0705, N1305, N0205) → angPoli13
  240. Angle(N0201, N0901, N0401) → angPoli14
  241. Angle(N0914, N1314, N1614) → angPoli15
  242. Vector(O, O + (Element[tetraL5, 1] * i + Element[tetraL5, 2] * j + Element[tetraL5, 3] * k) / 3) → zzVectorTetra5
  243. Segment(N1103', N1110') → n1'
  244. Segment(N0302', N0304') → n10'
  245. Segment(N0304', N0307') → n11'
  246. Segment(N0307', N0302') → n12'
  247. Segment(N1013', N1002') → n13'
  248. Segment(N1013', N1009') → n14'
  249. Segment(N1002', N1009') → n15'
  250. Segment(N1216', N1209') → n16'
  251. Segment(N1204', N1216') → n17'
  252. Segment(N1204', N1209') → n18'
  253. Segment(N1516', N1216') → n19'
  254. Segment(N1110', N1112') → n2'
  255. Segment(N0304, N1204) → n20
  256. Segment(N0304', N1204') → n20'
  257. Segment(N0302', N1002') → n21'
  258. Segment(N1209', N1009') → n22'
  259. Segment(N1513', N1013') → n23'
  260. Segment(N1507', N0307') → n24'
  261. Segment(N0708', N1608') → n25'
  262. Segment(N0408', N1608') → n26'
  263. Segment(N1608', N1614') → n27'
  264. Segment(N1614', N0914') → n28'
  265. Segment(N0705', N1305') → n29'
  266. Segment(N1103', N1112') → n3'
  267. Segment(N1314', N1614') → n30'
  268. Segment(N0705', N0205') → n31'
  269. Segment(N0205', N1305') → n32'
  270. Segment(N1305', N1314') → n33'
  271. Segment(N0705', N0708') → n34'
  272. Segment(N0708', N0408') → n35'
  273. Segment(N0205', N0201') → n36'
  274. Segment(N1314', N0914') → n37'
  275. Segment(N0408', N0401') → n38'
  276. Segment(N0201', N0401') → n39'
  277. Segment(N1103', N1115') → n4'
  278. Segment(N0201', N0901') → n40'
  279. Segment(N0401', N0901') → n41'
  280. Segment(N0901', N0914') → n42'
  281. Segment(N0506', N0806') → n43'
  282. Segment(N0506', N0106') → n44'
  283. Segment(N1406', N0506') → n45'
  284. Segment(N0806', N1406') → n46'
  285. Segment(N0806', N0106') → n47'
  286. Segment(N0106', N1406') → n48'
  287. Segment(N1115', N1112') → n5'
  288. Segment(N1115', N1110') → n6'
  289. Segment(N1516', N1513') → n7'
  290. Segment(N1513', N1507') → n8'
  291. Segment(N1507', N1516') → n9'
  292. Segment(CV6, CV11) → diagonal'
  293. Segment(G, J) → e
  294. Segment(K, L) → q
  295. Segment(K, N) → s
  296. Segment(N, Q) → a_1
  297. Segment(Q, L) → b_1
  298. Segment(K, R) → c_1
  299. Segment(R, S) → d_1
  300. Segment(S, N) → e_1
  301. Segment(S, U) → f_1
  302. Segment(U, Q) → g_1
  303. Segment(R, V) → h_1
  304. Segment(V, L) → i_1
  305. Segment(V, U) → j_1
  306. Segment(W, Z) → p_1
  307. Segment(W, A_1) → q_1
  308. Segment(A_1, B_1) → r_1
  309. Segment(Z, B_1) → s_1
  310. Segment(W, D_1) → t_1
  311. Segment(D_1, E_1) → a_2
  312. Segment(E_1, A_1) → b_2
  313. Segment(E_1, F_1) → c_2
  314. Segment(F_1, B_1) → d_2
  315. Segment(D_1, G_1) → e_2
  316. Segment(G_1, F_1) → f_2
  317. Segment(G_1, Z) → g_2
  318. Segment(N, A_1) → h_2
  319. Segment(S, E_1) → i_2
  320. Segment(W, K) → j_2
  321. Segment(D_1, R) → k_2
  322. Segment(G_1, V) → l_2
  323. Segment(Z, L) → m_2
  324. Segment(F_1, U) → n_2
  325. Segment(B_1, Q) → p_2
  326. Segment(J, H_1) → r_2
  327. Segment(H_1, I_1) → s_2
  328. Segment(I_1, J_1) → t_2
  329. Segment(J_1, K_1) → a_3
  330. Segment(K_1, L_1) → b_3
  331. Segment(L_1, M_1) → c_3
  332. Segment(M_1, N_1) → d_3
  333. Segment(N_1, O_1) → e_3
  334. Segment(O_1, P_1) → f_3
  335. Segment(P_1, Q_1) → g_3
  336. Segment(Q_1, R_1) → h_3
  337. Segment(R_1, S_1) → i_3
  338. Segment(S_1, T_1) → j_3
  339. Segment(T_1, V_1) → k_3
  340. Segment(V_1, W_1) → l_3
  341. Segment(W_1, Z_1) → m_3
  342. Segment(Z_1, A_2) → n_3
  343. Segment(A_2, K_2) → p_3
  344. Segment(L_2, K_2) → q_3
  345. Segment(L_2, M_2) → r_3
  346. Segment(M_2, N_2) → s_3
  347. Segment(N_2, O_2) → t_3
  348. Segment(O_2, P_2) → a_4
  349. Segment(P_2, Q_2) → b_4
  350. Segment(Q_2, R_2) → c_4
  351. Segment(R_2, S_2) → d_4
  352. Segment(S_2, T_2) → e_4
  353. Segment(T_2, U_2) → f_4
  354. Segment(U_2, V_2) → g_4
  355. Segment(V_2, W_2) → h_4
  356. Segment(W_2, Z_2) → i_4
  357. Segment(A_3, B_3) → j_4
  358. Segment(C_3, D_3) → k_4
  359. Segment(B_3, E_3) → l_4
  360. Segment(E_3, F_3) → m_4
  361. Segment(F_3, G_3) → n_4
  362. Segment(C_3, I_3) → q_4
  363. Segment(I_3, J_3) → r_4
  364. Segment(J_3, D_3) → s_4
  365. Segment(H_3, K_3) → p_4
  366. Segment(L_3, N_3) → t_4
  367. Angle(W3, W2, W4) → w1Ang
  368. Angle(W5, W2, W7) → w2Ang
  369. Angle(W6, W2, W5) → w3Ang
  370. Angle(W4, W1, W8) → w4Ang
  371. Angle(W5, W7, W6) → w5Ang
  372. Angle(W4, W7, W3) → w6Ang
  373. Segment(W1, W2) → w1A1
  374. Segment(W7, W3) → w1A10
  375. Segment(W6, W5) → w1A11
  376. Segment(W7, W8) → w1A12
  377. Segment(W2, W3) → w1A2
  378. Segment(W3, W4) → w1A3
  379. Segment(W4, W1) → w1A4
  380. Segment(W1, W6) → w1A5
  381. Segment(W6, W8) → w1A6
  382. Segment(W8, W4) → w1A7
  383. Segment(W2, W5) → w1A8
  384. Segment(W5, W7) → w1A9
  385. Join({{1, 1, -1}} * MM) → MW1
  386. Join({{1, 1, 1}} * MM) → MW2
  387. Join({{1, -1, 1}} * MM) → MW3
  388. Join({{1, -1, -1}} * MM) → MW4
  389. Join({{-1, 1, 1}} * MM) → MW5
  390. Join({{-1, 1, -1}} * MM) → MW6
  391. Join({{-1, -1, 1}} * MM) → MW7
  392. Join({{-1, -1, -1}} * MM) → MW8
  393. Segment(P, CW1) → wrayo1
  394. Segment(P, CW2) → wrayo2
  395. Segment(P, CW3) → wrayo3
  396. Segment(P, CW4) → wrayo4
  397. Segment(P, CW5) → wrayo5
  398. Segment(P, CW6) → wrayo6
  399. Segment(P, CW7) → wrayo7
  400. Segment(P, CW8) → wrayo8
  401. Segment(CW3, CW4) → a_5
  402. Segment(CW7, CW8) → b_5
  403. Segment(CW2, CW1) → c_5
  404. Segment(CW5, CW6) → d_5
  405. Segment(CW3, CW7) → e_5
  406. Segment(CW7, CW5) → f_5
  407. Segment(CW5, CW2) → g_5
  408. Segment(CW3, CW2) → h_5
  409. Segment(CW8, CW4) → i_5
  410. Segment(CW4, CW1) → j_5
  411. Segment(CW1, CW6) → k_5
  412. Segment(CW6, CW8) → l_5
  413. Segment(NW0304', NW0302') → n_5
  414. Segment(NW0307', NW0302') → p_5
  415. Segment(NW0307', NW0304') → q_5
  416. Segment(NW0307, NW0302) → r_5
  417. Segment(NW0302, NW0304) → s_5
  418. Segment(NW0307, NW0304) → t_5
  419. Max(tetraMW0, 1 - tetraMW0) → tetraMW
  420. Angle(NW0302, NW0304, NW0307) → angWPoli1
  421. Segment(NW0408, NW0401) → m_5
  422. Segment(NW0201, NW0205) → b_6
  423. Segment(NW0705, NW0708) → d_6
  424. Segment(NW0705', NW0708') → f_6
  425. Segment(NW0708', NW0408') → g_6
  426. Segment(NW0408', NW0401') → h_6
  427. Segment(NW0401', NW0201') → i_6
  428. Segment(NW0201', NW0205') → j_6
  429. Segment(NW0205', NW0705') → k_6
  430. Segment(NW0806, NW0506) → l_6
  431. Segment(NW0506, NW0106) → m_6
  432. Segment(NW0806, NW0106) → n_6
  433. Segment(NW0806', NW0506') → p_6
  434. Segment(NW0506', NW0106') → q_6
  435. Segment(NW0806', NW0106') → r_6
  436. Segment(W3, W6) → s_6
  437. Segment(NW0705, NW0205) → a_6
  438. Segment(NW0708, NW0408) → c_6
  439. Segment(NW0201, NW0401) → e_6
  440. Segment(CW3, CW6) → b_7
  441. Max(tetraM1, 1 - tetraM1) → tetraM
  442. Segment(NZ0302, NZ0304) → t_6
  443. Segment(NZ0302', NZ0304') → a_7
  444. Segment(NZ0401, NZ0201) → c_7
  445. Segment(NZ0401', NZ0201') → d_7
  446. Segment(W3, W1) → e_7
  447. Segment(CW3, CW1) → f_7
  448. Segment(NY0304', P) → g_7
  449. Segment(J_2, Q_3) → h_7
  450. Segment(Q_3, R_3) → i_7
  451. Segment(R_3, S_3) → j_7
  452. Segment(S_3, T_3) → k_7