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Monthly Archives: April 2016

Arc length is just:

L = rθ

Verify this by acknowledging

C = 2πr

In that case, θ = 2π

4d space is simply overlapping 3d volumes.

each dimension you add multiplies the previous dimension by infinitely many of the previous “type” of object.

For example:
0d: a point.
1d: a 1d line, or infinite stack of parallel 0d points
2d: a 2d plane, or infinite stack of parallel 1d lines
3d: a 3d volume, or infinite stack of parallel 2d planes
4d: 4d space, or “infinite stack” of parallel 3d volumes

parallel 3d volumes may seem hard to visualize, until you realize that they are just overlapping 3d spaces.

if you know computer graphics, think of rendering the same thing twice in the same spot. for example, a character, once with normal skin and once with werewolf skin. the second render introduces a “4th dimension” or a “parallel reality” for the character. if both renders are supposed to represent the same character, and these impassible objects can overlap, then we are now talking about parallel 3d spaces (since overlap of solid objects should not be possible in a single 3d space). the character now appears in 4d space, where he experiences the “parallel realities” of both being “normal” and “werewolf skinned” at the same time.

There’s an odd quirk in PowerPoint 2016 that causes the Text node in the Slide Master to not show up in the children.

  1. Insert Slide Master
  2. Move the text in the MASTER SLIDE off to the right, so you can clearly see which ones it is has skewed in of the LAYOUTS that are automatically set up below.
  3. Delete all the template layouts so you just have the Slide Master EXCEPT the 2nd Layout that has the movable Content node.
  4. You can duplicate this Layout node for additional nodes that have linked Content to the Master node.


This is the only Layout whose Content node modes with the Text in the Master. It’s difficult to get Powerpoint to behave this way otherwise.

By default, PowerPoint cleans up (by deletion) all unused master slide layouts almost immediately after you click “Close Master View”. As such, be sure to PRESERVE YOUR LAYOUTS by clicking the thumbtack icon just prior to closing out of the Slide Master View.