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Table 2 The 14 rays of the cone C23,45. Each ray is determined by a vector shown in the second column. The third column shows, for each ray, which cones it belongs to. If a cone is starred then the ray is on the boundary of that cone, but not a ray of it.

From: On the optimality of the neighbor-joining algorithm

Type rays Cones
I (-3, 5, -3, -1, 5, -3, -1, 1, 1, -1) (-3, 5, -3, -1, 1, 1, -1, 5, -3, -1) (5, -3, -3, -1, -3, 5, -1, 1, 1, -1) (1, 1, -3, -1, -3, 5, -1, 5, -3, -1) (5, -3, -3, -1, 1, 1, -1, -3, 5, -1) (1, 1, -3, -1, 5, -3, -1, -3, 5, -1) C23,45, C23,15, C23,14, C 12 , 34 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4qam0aa0baaSqaaiabigdaXiabikdaYiabcYcaSiabiodaZiabisda0aqaaiabgEHiQaaaaaa@32AC@ , C 34 , 12 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4qam0aa0baaSqaaiabiodaZiabisda0iabcYcaSiabigdaXiabikdaYaqaaiabgEHiQaaaaaa@32AC@ C23,45, C23,15, C23,14, C 12 , 35 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4qam0aa0baaSqaaiabigdaXiabikdaYiabcYcaSiabiodaZiabiwda1aqaaiabgEHiQaaaaaa@32AE@ , C 35 , 12 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4qam0aa0baaSqaaiabiodaZiabiwda1iabcYcaSiabigdaXiabikdaYaqaaiabgEHiQaaaaaa@32AE@ C23,45, C23,15, C23,14, C 24 , 13 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4qam0aa0baaSqaaiabikdaYiabisda0iabcYcaSiabigdaXiabiodaZaqaaiabgEHiQaaaaaa@32AC@ , C 13 , 24 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4qam0aa0baaSqaaiabigdaXiabiodaZiabcYcaSiabikdaYiabisda0aqaaiabgEHiQaaaaaa@32AC@ C23,45, C23,15, C23,14, C 25 , 13 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4qam0aa0baaSqaaiabikdaYiabiwda1iabcYcaSiabigdaXiabiodaZaqaaiabgEHiQaaaaaa@32AE@ , C 25 , 13 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4qam0aa0baaSqaaiabikdaYiabiwda1iabcYcaSiabigdaXiabiodaZaqaaiabgEHiQaaaaaa@32AE@ C23,45, C23,15, C23,14, C 24 , 35 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4qam0aa0baaSqaaiabikdaYiabisda0iabcYcaSiabiodaZiabiwda1aqaaiabgEHiQaaaaaa@32B4@ , C 35 , 24 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4qam0aa0baaSqaaiabiodaZiabiwda1iabcYcaSiabikdaYiabisda0aqaaiabgEHiQaaaaaa@32B4@ C23,45, C23,15, C23,14, C 25 , 34 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4qam0aa0baaSqaaiabikdaYiabiwda1iabcYcaSiabiodaZiabisda0aqaaiabgEHiQaaaaaa@32B4@ , C 25 , 34 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4qam0aa0baaSqaaiabikdaYiabiwda1iabcYcaSiabiodaZiabisda0aqaaiabgEHiQaaaaaa@32B4@
II (-1, 1, -1, 1, 1, -1, -1, 1, 1, -1) (-1, 1, -1, -1, 1, 1, 1, 1, -1, -1) (1, 1, -1, -1, -1, 1, -1, 1, -1, 1) (1, -1, -1, 1, -1, 1, -1, 1, 1, -1) C12,45, C12,34, C23,45, C23,15, C34,15, C34,12, C45,23, C45,12, C15,34, C15,23 C12,45, C12,35, C23,45, C23,14, C35,14, C35,12, C45,23, C45,12, C14,35, C14,23 C25,14, C25,13, C23,14, C23,45, C13,45, C13,25, C14,23, C14,25, C45,13, C45,23 C24,15, C24,13, C23,15, C23,45, C13,45, C13,24, C15,23, C15,24, C45,13, C45,23
III (1, -1, -1, 1, 1, -1, -1, -1, 3, -1) (1, -1, -1, -1, -1, 3, 1, 1, -1, -1) (1, -1, -1, 1, 1, -1, -1, -1, 3, -1) (1, -1, -1, -1, -1, 3, 1, 1, -1, -1) C23,45, C23,15, C12,45, C12,35, C24,15, C24,35, C35,24, C35,12, C15,24, C15,23, C45,12, C45,23 C23,45, C23,14, C12,45, C12,34, C25,14, C25,34, C34,25, C34,12, C14,25, C14,23, C45,12, C45,23 C23,45, C23,15, C13,45, C13,25, C34,15, C34,25, C25,34, C25,13, C15,34, C15,23, C45,13, C45,23 C23,45, C23,14, C13,45, C13,24, C35,14, C35,24, C24,35, C24,13, C14,35, C14,23, C45,13, C45,23