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Butterflies, like many other insects, understand that I am getting near from behind. How do they feel this - through vision, hearing, or some other sense?

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Short answer
Butterflies can see behind them.

Background
Merry et al. (2006) have estimated the field of view of the butterfly, by investigating the Orange Sulphur butterfly (Colias eurytheme):

Orange Suplhur
Colias eurytheme. Source: Massachusetts Butterfly Club

The authors found that this butterfly has a very large visual field, encompassing 93% of the spherical space around the animal. There was a small blind spot behind the head not seen by either eye. Another study by the same group showed similar results in another species of butterfly, the Empress Leilia (Asterocampa leilia). Here is a picture of the estimated binocular field of view of the Orange Sulphur, revealing that the animals' vision is virtually spherical, with only a small blind spot at the back (the white area):
FieldofView
Source: Merry et al., 2006. Grey is the field of view, white the blind spot posteriorly. Cardinal directions are indicated (Dorsal, Ventral, Posterior, Left and Right).

Here are a few pictures of other species of butterfly that show the protruding compound eyes covering a large field of view:

enter image description here enter image description here
Sources: Alex Sukonkin and PhotographersDirect

The feature of large visual fields in insects is, however, not restricted to butterflies (Lutowski & Warrant, 2002). Another insect I often try sneak upon to take pictures are dragon flies, and more often than not they fly away regardless of the angle I approach them. They too have these large compound eyes with a field of view extending to the back:

dragon fly
Source: Nature North


References
- Lutowski & Warrant, J Comp Physiol 2002;188:1-12
- Merry et al., J Insect Physiol 2006;52:240–8

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Butterflies have nearly 360 degree field of vision (see e.g. Belth. Butterflies of Indiana), so they can see you or other predators approaching from behind at the same time as they examine the flower they are feeding from. A large field of vision is common in many animals that are preyed upon, and for instance many birds (e.g. waders) have 360 vision, some even with overlapping, binocular vision behind their heads.

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  • $\begingroup$ You beat me by 21 seconds! +1 for your evolutionary perspective :) $\endgroup$ – AliceD Mar 16 '15 at 10:07
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    $\begingroup$ @AliceD Well, your answer is more complete with better references though. $\endgroup$ – fileunderwater Mar 17 '15 at 9:42

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