About Geom-e-Tree™ and geom-e-trees

Changing the Branching Factor

To change the number of branches at each node, use a two-finger tap on the upper or lower half of display — two fingers, one tap. This is a change from Geom-e-Tree 1.2.

• Two-Finger Tap in the ^upper half to increase the branching factor by 1.
• Two-Finger Tap in the _lower half to decrease the branching factor by 1.

The branching factor can be increased to 9 or decreased to 2 in this way.

Varying the Angle by Panning

To continuously vary the angle between branches, move one finger up or down on the display. This is called panning.

All angles in the Tree will change as you pan.

• Start anywhere — Pan up or down, and the angles will respond.
• Pan repeatedly (by lifting your finger) to change the angles a lot.

To change the Tree angle exactly one degree, use a single-finger tap.

• Tap in the ^upper half of the display to close the angles by 1°.
• Tap in the _lower half of the display to open the angles 1°.

Angles stop opening up at 359° and stop closing at 0°.

Varying the Common Ratio by Pinching and Spreading

To change the common ratio between the levels of branches, place two fingers on the display and either bring them together or spread them apart.

• Pinching a Tree generally results in complex patterns of overlapping branches. Think compression.
• Spreading a Tree generally makes a simpler pattern, one with less overlapping. Think rarefaction.

Red prohibition signs on the display indicate that you can't pinch the common ratio any further.

Tapping one finger twice in quick succession anywhere on the display immediately sets the common ratio so the branches will be as long as possible without overlapping each other (in normal conditions). A green border will appear when the Tree is in this mode.

The Tree will stay in this mode until you double-tap again, or pinch or spread to take manual control of the common ratio. See the FAQ on our web site to learn more about this special automatic mode.

Playing with the Demo Tree theme

Check the Tree tab, at the bottom left of your iPad or iPhone display to see if it has this 2-branched Demo Tree. The Demo Tree is intentionally simple so you can see the effect of your gestures.

If the current Tree doesn't look like a Demo Tree, switch to the Themes tab and scroll down to the Demo Tree theme. Tap on Demo Tree. The current Tree will have the Demo Tree theme applied to it. Now play. Alternatively, you can find this simple Tree in the Arboretum. Double Tap on it and you should be good to go.

• Vary the angle between branches by panning up and down.
• Vary the ratio between the levels by pinching and spreading.
• Change the branching factor by using two-finger taps.
• Double Tap. Watch the Tree auto-determine the ratio as you change the angle.
• Try changing the Tree somehow, intentionally. Verbalize what you are doing and seeing.
• Play for a while, then explore some other theme(s).
• If you noticed the Demo Tree doing some strange things, see the next section!

Wondering what the Tree is Doing

Did you sometimes see the number of levels change when you changed the number of branches?

Explanation: The Demo Tree theme is limited to 60 lines for simplicity.
The table shows the results of this constraint.

In general, changing the branching factor always changes the Tree dramatically (even if the number of levels stays the same). The change is instantaneous.

Did the whole Tree seem to move up and down? Did the length of the trunk change as you gestured?

Explanation: Trees are dynamically scaled to fit on the display. Sometimes it's the width of the Tree that matters, other times it's the height. If this auto-scaling weren't done, you wouldn't see parts of the Tree — they'd be off-display! The Tree is also automatically centered as it changes. These things account for the movement.

Did you notice when varying the angle that some branches moved up, while others moved down?

Explanation: Try this - Start with a 5+ branched Tree, open the angle by panning down, down, down… open the angle a lot. As the Tree opens, you'll see some branches moving clockwise and others moving counter-clockwise. This results in the confusion — the branches are just passing each other on their way around.

Gesturing with Intention

What to do:

• When pinching or spreading, be sure both your thumb and index finger make contact with the display at the same time, otherwise the Tree will only respond to one finger.
• For finer control over pinching and spreading, start with your fingers farther apart. You can move one finger or both in order to gesture. You can also use one finger from each hand for maximum spread.
• Geom-e-Tree doesn't respond well to wild gestures. Relax. You will have a more enjoyable experience if you let the Tree respond to your gestures — it needs more time to respond when its branches are hyper extended or it's casting long shadows.
• If you get lost, just double tap with one finger and the Tree will probably go back to a simpler, less overlapping pattern. The angle won't be changed. If the Tree doesn't change, then it's probably already in the Perfect Common Ratio Mode. (PCR Mode is indicated by a green border around the display.) You'll just have to unwind the Tree angle, or start over with a fresh Tree from the Arboretum until you can figure out what part of the forest you got lost in. Don't give up — geom-e-trees can be tricky! Consider checking the FAQ for help.

What not to do:

• A side-to-side gesture doesn't mean anything. The Tree may kinda look like it's doing something, but only the up-down component of a panning gesture is used.
• You may think that touching the Tree itself in some way does something. Nope. Only gestures on the display as described are meaningful.
• When single-tapping through a number of degrees, don't tap too quickly or you'll be double-tapping. See Setting and Unsetting the Perfect Common Ratio Mode.

Watching Geom-e-Tree in Action on YouTube

Check out the 5 minute video Geom-e-Tree in Action on YouTube. The video goes through the different gestures or Moves and offers a couple tips. If you have read this Help, you shouldn't need the Video. But, if you are having some kind of problem, for example you can't get a particular pose you've seen, or whatever, watching the video may give you an Aha! A link is on the Videos page of our web site. (Be aware of using cellular data — WiFi is recommended.)

Visiting the Arboretum

In addition to the eight permanent Reference Trees, you'll find a few interesting ones to vivify. Once the Tree is live, try changing the angle or ratio for a surprise.

The Arboretum is described fully in its own section below.

The Unit Length Tree Tour

Before the tour… Note that a Common Ratio of 1.0 means that ALL branches will be the same length -- even though they are increasing in number at every level. At special angles, the branches align to form hexagonal, square, and isometric grids!

The tour: Start with the binary Reference Tree from the Arboretum. (It's at 180° with a common ratio of ~1.42). Pinch it down until the common ratio is 1.0. You should see a simple square grid form. Now, pan up to vary the angle and you'll see the Hexagonal grid at 120°. Go back the other way (pan down), past the square grid you saw at 180° and you'll come to an inverted isometric grid at 240°. Side Trips: Pinch any grid all the way down to take a look, then spread to come back to the grid and continue your tour.

You can make similar tours with higher order Trees (i.e. higher branching factors).

The Doubling Tree Tour

Before the tour… A Common Ratio of 0.5 means that branches will DOUBLE in length from one level to the next in the Tree. As you vary the angle, the branches have to go somewhere. As it turns out, they form some simple, but deeply complex patterns.

The tour: Vivify the ternary (3-branched) Tree from the Arboretum. The angle between its branches is 120°. Pinch it all the way down until a red prohibition indicator appears. The Common Ratio is now 0.5. You can confirm that by going to the Share/Save tab. You should be seeing an equilateral triangle pose! Now decrease the angle slowly (pan up) and you'll see a parade of polygon poses at 90°, 72°, 60°, 45°, 36°…

You would not believe us if we told you what you will see at 144°. To our knowledge, this was unknown prior to its discovery with Geom-e-Tree. Check it out.

Different Themes show Different Things

Color can be used for more than just making pretty patterns. The rainbow themes make it easy to see what level branches are on, no matter what pose a Tree is in. The black & white themes show even-odd levels. The Branching Order themes show left vs right branches in binary trees, left-middle-right branches in ternary trees, and so on. Colored Node themes (Fundamental Particles, Juggler) show the level of all Tree nodes regardless of their location. Each theme tells us something a little different about geom-e-tree geometry. Check them out!

Existing vs New Arboretums

If you got Geom-e-Tree 1.3 as an update, your existing Arboretum was updated to include Reference Trees for 8 and 9-branched geom-e-trees, and some removable sample Trees were added.

If your Geom-e-Tree App was freshly installed, its Arboretum will have Reference Trees for 2 thru 9-branched geom-e-trees. New Arboretums will also have a few removable sample Trees to play with.

See What's New in this Version.

Branching Order Themes

The first new kind of theme shows the branching order of the Tree. In the simplest case, a binary tree's left branches are white and its right branches are black. For higher order trees (branching factor > 2), we number the branches at each node 1, 2, 3… and assign colors to 1, 2, 3… so the branches are colored based on their left-to-right order at the node they branch from. This theme shows how the structure can curl around and fold up.

Allometric Line Themes

Also in this version we introduce the notion of Allometry (scaling) by changing the thickness of the branches as well as their length as the common ratio is varied. These themes are quite striking, but it's impossible to keep an allometric-themed Tree from swelling up and engulfing the display at extreme ratios, so they are more constrained than other themes. It's best to think of these as 'beta' versions of these themes. See our FAQ.

Dots at Nodes

For something completely different, we have dot-based themes that feature the nodes of the Tree rather than branches. The underlying geometry is the same — the graphic rendition is different.

These themes are incredible. You may see all the dots merge into just a few piles, or arrange themselves in groups. When you move away from those coincidences, the groupings explode and the dots move on their way to some other such arrangement / grouping. A variant theme (Fundamental Particles) colors the dots based on their level to produce patterns that beg explanation.

Another 'beta' theme combines Dots with Allometry where structure seemingly materializes out of thin air as you pinch the display. Other themes (Circles, Wafers, and Rings) use the geometry of circles centered at the midpoint of each branch, with a diameter equal to the length of the branch!

Dots and Bonds

The most fantastic new theme (Dendritic Group Theory) combines plain dots with line segments or bonds at each node. Each node appears to point to the other nodes it's connected to. The bonds are color-coded by level, so you see nodes grouping together in various colored clusters, and zipping around to regroup as you vary the angle and ratio. It's as fun to watch as it is beautiful.

Diamonds and Lenses

The diamond and the lens are basic geometric-axial shapes that can represent the branches of a geom-e-tree. A lens is a lamina of two spherical arcs. We use symmetric lenses with a 60° arc. Our diamond is a rhombus of two 45° or 60° triangles. See Rhombus, Lozenge and Vesica Piscis on Wikipedia.

More Help & Fun

On the internet...

• A very helpful FAQ is on geom-e-tree.com.
• Some wild Geom-e-Trees are on our Explore page (~2MB).
• Our demo Geom-e-Tree in Action is on YouTube. See Videos.
• The original Geom-e-Tree (1980) animation is on YouTube. See Videos.
• Write to us: support@geom-e-tree.com. Your feedback is welcome.
• A printable Gesture Summary and Reference Sheet is also available on our web site.
• This same Help page is available on our web site for your reference. You can open it in Safari for printing.

Time Haven Media

At the time of this release, we have three other apps in the App Store, and a fifth on the way:

• Geom-e-Twee (Twee) is the children's version of Geom-e-Tree. Twee has a dozen simple themes and draws Twees with 2 or 3 branches. It does not have an Arboretum.
• PolygonFlux (Flux) is based on Polygonal Billiards, an active area of geometric research. Flux generates some very mysterious patterns. In simplest terms, it traces the path of a point bouncing around inside a regular polygon of your choice. We like to say that PolygonFlux lets you experience the flow of energy inside a polygon.
• PolygonTrix (Trix) is a Math Art Game based on PolygonFlux. It's a digital arcade of angles and energy that lets you explore the Space-Time dynamics hidden in polygons. PolygonTrix is NOT a shooting game — it's an amusing geometric puzzler with a particle physics twist.
• PolygonJazz (Jazz) combines musical notes with billiards. Look for Jazz in January 2014 and Listen to the Polygons.

We will release MacOS versions of Tree and Flux as soon as we can. You can always find our current apps and other news on our mothership site: Time Haven Media Company.

Geometry may be a Universal Language...

...but please forgive our lack of Internationalization. We appreciate the friends we have in France, Germany, Spain, Italy, Japan, and other countries. Someday we hope to be able to afford professional translation of this Help and the FAQ into five or more languages and have those appear directly in Geom-e-Tree. Until then, we have provided links on our support page for Google translations.

See translations for French, German, Spanish, Italian, Japanese, etc.

Keeping the Trees Free

All simple, plain-color fine-lined patterns produced by Geom-e-Tree are natural mathematical forms, and therefore are in the public domain. We only ask that any commercial or artistic use of a geom-e-tree produced by Geom-e-Tree adhere to the Creative Commons Attribution License Conditions, stating clearly that the pattern came from the Geom-e-Tree iPhone or iPad application. Please see Creative Commons About Licenses.

A Challenge for other Developers

The Geom-e-Tree™ and Geom-e-Twee™ apps themselves are ©2013, by John Miller, Portland, Oregon, USA, and their app names are trademarked. These apps will have features added in later versions. We really don't want to see copycat variants of our basic design. We ask other developers to respect our right to improve Geom-e-Twee and Geom-e-Tree, and challenge them to come up with their own creative ideas to enrich the electronic playground and classroom. Thanks!