Steps to Reproduce

Create meshes of identical size with differing numbers of faces. To make the issue obvious the Z dimension should be dominant. I have pre-created assets on aditi whcih are the ones shown in the images of the blog linked (http://beqsother.blogspot.co.uk/2016/10/bug-hunting-fixing-ancient-lod-issue.html)

1. face Aditi asset UUID
   1 face 1d71cddb-59c5-4555-7789-7ffbdd7330a1
   2 face b512d9ba-7c40-13b5-2911-db561c728f5f
   3 face d3e37524-c53f-d4f2-328c-0971bf645521
   4 face cf5f8206-9de5-dc28-06a5-f544f285d8de
   5 face 6c60ffd2-baba-c52b-5a76-1b67bdbc0f9a
   6 face bcfbf6d5-047a-d5a2-2a67-e5966921ef13
   7 face fa6b61c0-5e7a-fc38-1435-cc1692152a4e
   8 face 2fbd426d-9e87-2b34-d388-336b89d25c73

Line them all up and (optionally) with your LOD factor set to 1 to make the LOD shift happen sooner pull the camera back. The meshes with 3 and 4 faces should crumple immediately followed by the 5,6,7 and 8 face meshes that will crumple slightly earlier than the remaining 1 and 2 face meshes.
https://gyazo.com/63f3a72cfcf3570528913efc0885ac94

There are 3 different results for the LODBias depending on the number of material faces, as this is clearly arbitrary and has nothing to do with geometry it must be considered a bug. The table of results by material count is as follows.

1. faces BIAS
   1 <0.6,0.6,0.6>
   2 <0.6,0.6,0.6>
   3 <0.6,0.6,0.0>
   4 <0.6,0.6,0.0>
   5 <0.5,0.5,0.5>
   6 <0.5,0.5,0.5>
   7 <0.5,0.5,0.5>
   8 <0.5,0.5,0.5>

Actual Behavior

First reported in 2011, a bug exists that results in different (and generally incorrect) LOD radius being used for Meshes depending upon the number of material faces. I have linked a number of the associated reports some of which are no longer accessible to non-Lindens.

A recent example of the bug can be found documented in my blog from July http://beqsother.blogspot.co.uk/2016/07/tell-your-friends-old-bug-that-people.html
Visually is is best seen in this image
https://i.gyazo.com/690d132247bc8ac904600126bedd2233.gif

Expected Behavior

As the LOD is dictated by the bounding box of the object, and not any other arbirtary detail, all mesh objects of the same scale should LOD identically.
Arguably the correct bias is that shown for 5,6,7,8 as this would give half of the long diagonal of the object bounding box.

Other information

I have a fix available but I cannot yet submit until https://jira.secondlife.com/browse/BUG-40625 has been resolved and I have completed the form.
In the fix I force all mesh to have a bias equivalent to the existing behaviour for 1 and 2 material meshes as this ensures that no Mesh is adversely affected. However I suspect that strictly correct behaviour is to move everything to the 0.5 all axis bias seen on 5,6,7 & 8 material face meshes.

You can read details of the fix in my blog. http://beqsother.blogspot.co.uk/2016/10/bug-hunting-fixing-ancient-lod-issue.html