Correction to ‘Stiffest elastic networks’

Proc. R. Soc. A 470, 20130611. (Published 12 February 2014) (doi:10.1098/rspa.2013.0611)

In the calculations performed for the Union Jack lattice p. 14 to find the ratio of beam cross-sections that fulfill the isotropy conditions (4.3), we mistyped some intermediate formulae (although the final result is correct):

• — In the second paragraph, last sentence:

  • ‘the only restrictive condition is 〈cos 4θ〉 = 0, yielding 2(l₁ s₁ − l₂ s₂) = 0’
  • should be replaced by ‘the only restrictive condition is 〈cos 4θ〉 = 0, yielding 2l₁ s₁ − l₂ s₂ = 0’
• — and:

  • ‘As, from geometry, $l_2=l_1/\sqrt{2}$’
  • should be replaced by ‘As, from geometry, $l_2=l_1\sqrt{2}$’

This is in agreement with figure 6a.

Moreover, we realized that the average introduced p. 8 and defined as: $⟨q⟩=\sum _{(i,j)}{v}_{ij}{q}_{ij}/\sum _{(i,j)}{v}_{ij}$, where the sum is carried over all beams (i, j) is properly defined only if q_ij is a non-oriented quantity: q_ij = q_ji. Fortunately, throughout the paper, all the averaged quantities satisfy this property. We thus propose the following change:

• — p. 8, second paragraph:

  • ‘where the brackets denote the average defined, for any quantity q_ij, as $⟨q⟩=\sum _{(i,j)}{v}_{ij}{q}_{ij}/\sum _{(i,j)}{v}_{ij}$’
  • should be replaced by ‘where the brackets denote the average defined, for any quantity q_ij, as $⟨q⟩=\sum _{(i,j)}{v}_{ij}{q}_{ij}/\sum _{(i,j)}{v}_{ij}$. Note that this average assumes that q_ij = q_ji, which is always the case in the following’.