Eye structure and geometry defines resolution and costs. a) Fused rhabdom apposition eye, photoreceptors coding a pixel form fused rhabdom and send axons to a single neural module; b) neural superposition (NS) apposition eye, photoreceptor forms its own rhabdomere, photoreceptors with same optical axes code single pixel and send axons to single neural module; c) simple eye, as in a camera each photoreceptor codes a single pixel. d) Gradient of investment in spatial acuity: apposition eye, honeybee drone, Apis mellifera. From dorsal to ventral, D increases and Δϕ decreases to increase spatial resolution, rhabdom length,L, increases to increase SNRph. 10 µm thick longitudinal section, (DA) dorsal eye area (VA) ventral area. BM - retina’s basement membrane; E – equator separating dorsal and ventral regions. From Menzel et al., 1991, original micrograph, courtesy of Doekele Stavenga. e) Schematic section of locally spherical apposition eye region. Volumes of optics and photoreceptor array are determined by dimensions that constrain the quality of the spatial image coded by photoreceptors: lens diameter D, focal distance f′, interommatidial angle Δϕ = D/R where R is eye radius, and rhabdom(ere) length L.

The effect of trading investment in photoreceptor array for investment in dioptric apparatus. Schematic shows two eye regions of equal specific volume, the left investing more heavily in photoreceptors, the right more heavily on optics. The images they capture show that transferring resources from photoreceptor array to optics increases image sharpness and contrast at the expense of increasing noise.

The performance surface H(D, L) describes how information capacity changes across all geometrically permissible configurations of an eye of fixed cost - the eye’s morphospace. Red area: high-performance zone in which capacity is > 95% maximum. Surfaces plotted for model fly neural superposition (NS) eyes, with F-number, F = 2 and total cost, Ctot = 4 × 109 µm3 sr−1 at 3 values of photoreceptor energy tariff KE. Acceptance angle Δρ calculated using Snyder’s (1979) CoG approximation.

When model fly NS eyes are optimised for information capacity, eye structure, eye performance and the division of resources between optics and photoreceptor array depend on total investment, Ctot and photoreceptor energy tariff, KE. a) lens diameter D; b) interommatidial angle, Δϕ; c) rhabdomere length, L. Key in c) applies to a) and b). d) information capacity H in the region where plot of L vs Ctot at KE = 0.12 jumps. e) %Ctot allocated to photoreceptor array; f) photoreceptor energy cost as %Ctot. CoG, acceptance angle approximated by convolving Gaussians (Snyder, 1979); WOM, acceptance angle approximated according to wave optics model (Stavenga, 2004).

Measurements of Dipteran neural superposition (NS) eyes, extracted from published sources, 1. Land 1997; 2. Hardie 1985; 3. Stavenga et al. 1990; 4. Stavenga 2003a; 5. Land and Eckert 1985 ; 6. Stavenga 2003b; 7. Gonzalez Bellido et al. 2011; 8. Zeil 1983; 9. Wardill et al. 2017. Further details of the measurements used and results obtained are given in supplementary table S1.

Parameters defining the configurations of 12 fly NS eye regions, taken from 7 species (Table 1), compared to model NS eyes optimised for information capacity at given values of photoreceptor energy tariff KE, and total specific volume Vtot. a) lens diameter D, b) interommatidial angle Δϕ and c) eye parameter p = DΔϕ (keys as in a), d) rhabdomere length L. e) Log:log plot of L vs D, dashed line shows slope for isomorphic scaling, L ∝ D;. f) Specific volume of photoreceptor array, Vph expressed as % total specific volume of eye, %Vtot. CoG - acceptance angle approximated by convolving Gaussians (Snyder, 1979).

Measurements of fused-rhabdom apposition eyes, extracted from published sources, 1. Rossel 1979; 2. Labhart and Nilsson, 1995; 3. Menzel et al 1991; 4. Varela and Wiitanen, 1970; 5. Kelber and Somanathan, 2019. Details of measurements are given in supplementary table S1.

Theoretical results from apposition eye models optimised for information capacity compared to empirical data from 16 apposition eye regions taken from 3 species (Table 2). a) D vs specific volume Vtot; b) Δϕ vs Vtot c) L vs Vtot; d) photoreceptor array specific volume Vph as %Vtot vs Vtot. (e & f) Models demonstrate impact of photoreceptor costs and its dependence on total cost, Ctot. e) Photoreceptor cost, Cph, as %Ctot; f) Photoreceptor energy cost as %Cph. Models run with F-number F = 5.5, using COG approximation for acceptance angle (Snyder, 1979) and values of energy tariff KE given in keys.

Resource allocation in model simple eyes optimised for information capacity compared to optimised model neural superposition apposition eyes (NS). a) Schematic showing apposition eye and simple eye with identical spatial resolution, as defined by lens diameter, D, focal distance f′, rhabdom(ere) diameter drh and rhabdom(ere) length, L (after Kirschfeld, 1976). Note denser packing of photoreceptors inside the simple eye. (b - e) Properties of simple and NS eye models optimised for information capacity in full daylight. b) Photoreceptor length, L vs specific volume Vtot; c) photoreceptor specific volume Vph expressed as %Vtot vs Vtot. d) Photoreceptor investment Cph as %Ctot vs Ctot. e) photoreceptor energy cost as %Ctot vs Ctot. Models use Snyder’s (1979) CoG approximation for acceptance angle.

Information rate, H v.s. total cost Ctot for simple and apposition model eyes optimised to maximise information capacity in full daylight. a) Rates are higher in simple eyes and more sensitive to photoreceptor energy tariff, KE b) Efficiency, bits per unit specific volume per second, falls less steeply with increasing Ctot in simple eyes. Models use Snyder’s CoG approximation (1979) for acceptance angle.

Dependence of energy surcharge KE on time spent flying TF and hours of daylight DL, calculated for blowfly.