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If I have multiple objects selected can I use scaling to make the objects closer together (without rescaling the objects)?

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Yes, enable the pivot-point-align option.

This transforms the center points only.

One way to think of this is as a ferris-wheel - where each gondola stays in place, but you can rotate their locations.

Scale: scale

Rotate: Even though you didn't ask about this, it helps to understand how the feature works. rotate

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    $\begingroup$ That is awesome! I never realized this feature existed! +1. $\endgroup$ – CharlesL Jun 23 '13 at 18:09
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    $\begingroup$ I've just had a look in the Manual but there seems to be nothing about this useful little feature. $\endgroup$ – SteveW Jun 23 '13 at 22:51
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    $\begingroup$ Elegant and effective combination of simple tools is pure joy. $\endgroup$ – Haunt_House Aug 14 '13 at 23:47
  • $\begingroup$ Wait, did they change the name of that button? I swear the popup calls it "object centers only". Grawr, computer not handy. $\endgroup$ – StarWeaver Nov 27 '14 at 10:38
  • $\begingroup$ This does not work in edit mode, just a reminder. $\endgroup$ – John Leidegren Sep 10 '16 at 14:08
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As answered by @ideasman42, pivot point align option is a really awesome transformation tool. That coupled with Proportional Editing (Object Mode with Constant Falloff) the following transformations can be easily achieved.

Initial state of the objects

  1. Enable Pivot Point Align (Alt+,).
  2. Enable Proportional Editing (O).
  3. Choose Constant Falloff (Shift+O until constant falloff is displayed).

Enable Pivot Point Align, Proportional Editing with Constant Falloff

Left most object selected and scaled

Start scaling the objects and move the mouse while adjusting the proportional inflience area by scrolling the mouse wheel

Start scaling the objects

Middle object selected and scaled

In this case, the objects are proportionally scaled about the middle object.

Another way of scaling the same set of objects

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