About Geom-e-Tree™ and geom-e-trees

→ For those who are Updating to Geom-e-Tree Version 1.4 ←

Version 1.4 adds seven new themes based on diamond and lens shaped branches, generating beautiful tesselations. Version 1.3 doubled the number of themes found in Version 1.2.

Version 1.3 made the branching factor easier to change — A two-finger tap in the upper or lower half of the display increases or decreases the number of branches at each node in the Tree.

Starting with version 1.3, a double-tap toggles the Perfect Common Ratio Mode on or off.

See What's New below.

What's a geom-e-tree?

geom-e-tree diagram. From its trunk up through its branches, a geom-e-tree — or Tree — is a geometric figure that has the same structure repeated throughout:

As a special case, when the common ratio is 1/1, the trunk, branches, and twigs are all the same length!

Trees have many surprises. Geom-e-Tree 1.4, The App, provides an entertaining, educational, immersive experience of geom-e-trees.

How do I Use this App?

The first three sections explain how to use Geom-e-Tree — from using the basic gestures to really getting into it. To learn How to use this App, play with a Demo Tree while reading the first and second sections. When you're done with the demo, use a theme like Geometric Lite or Green Phosphor when following Help.

Here are some navigation aids for you return trips:

As you gain familiarity with Geom-e-Tree, remember to review this Help from time to time. Your learning will progress through how to control a Tree using gestures, through gaining the ability to explore Tree space without getting lost, to Seeing the Forest. We invite you to relax and enjoy all the brain-stimulating patterns Geom-e-Tree has to offer.

Controlling and Exploring Trees with Gestures

Geom-e-Tree responds to simple gestures. The entire display area can be used while gesturing. Geom-e-Tree has a special feature that can be toggled on/off with a gesture.

Changing the Branching Factor

Graphic depicting Two-Finger Tap 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.

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

Varying the Angle by Panning

Graphic depicting Tapping Gesture Graphic depicting Panning Gesture 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.

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

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

Varying the Common Ratio by Pinching and Spreading

Graphic depicting Spreading Gesture 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.

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

Setting and Unsetting the Perfect Common Ratio Mode

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.

Getting Started

This section has some exercises to get you familiar with Geom-e-Tree — to get you to think of it as a live object that responds to your gestures.

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.

geom-e-tree with Demo Tree theme 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.

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.

Demo Tree Branching Factor 23456789
Highest Level with fewer than 60 Lines Total 54333322
Total Number of Lines for Complete Level 314021314357 910

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.

Using Gestures Effectively

Now that you have the seen the basic gestures, learn how to use them effectively so you can play with the Tree.

Gesturing with Intention

What to do:

What not to do:

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.

Really Getting into Geom-e-Tree

At this point, you may understand what each gesture does, but you are probably dizzy from all the Trees you've seen. You may wonder how to get back to a Tree you saw, or how all the crazy patterns relate to one another. Consider this:

Unless you change the branching factor, a Tree simply morphs from one pose to another as you gesture.

Consider taking two tours, much like a knight tours a chessboard, to visit some special poses. Note: On both tours, you may need to single tap to adjust the angle in 1° steps in order to land on a certain pose.

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!

Hexagonal Grid Pose seen on Unit Length Tour 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.

Equilateral Triangle Pose seen on Unit Length Tour 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!

Sharing & Saving Geom-e-Trees

When a particular Tree strikes your fancy, go to the Share/Save tab… This tab shows what theme, branching factor, angle, and common ratio was used to make the Tree. Three buttons are provided:

Tip: You can switch to the Share/Save tab just to protect your current Tree from accidental change.

Feel free to share a Tree on social media or math-related site where appropriate. It's OK to use a Tree for a derivative work (Needlepoint for example). If possible, please refer to Geom-e-Tree so people know where it came from.

See Keeping the Trees Free below.

The Arboretum

The Arboretum is where you can collect and manage geom-e-trees that you like. Everything about the Tree is saved, including whether it was in the Perfect Common Ratio Mode. You can examine, vivify (activate), or remove a selected Tree in the Arboretum.

Arboretum Trash Can You can vivify a Tree, morph it however you want, and then save that new Tree back into the Arboretum. If you want to change a previously saved Tree — vivify it, adjust it, change the theme, etc, and then add that Tree to the Arboretum. If you no longer want the original, go to the Arboretum, single-tap the original, and then remove it.

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.

Geom-e-Tree Themes

Themes are 'special effects' you can choose from the Themes tab at the bottom of the display. All effects are based on the same Tree geometry.

The original version of Geom-e-Tree had plain black-lined Trees. The next version introduced themes with basically two coloring schemes for the lines — either a rainbow/spectrum theme or black & white theme for odd/even levels.

In this version we introduce several new kinds of themes.

Branching Order Themes

Traversal Order theme 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

Allometric Lines theme 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.

Allometric Dots theme 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

Dendritic Group Theory theme 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

Diamond and Lens themes 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.

What's New in Version 1.3 and 1.4?

Briefly, we changed two gestures, and doubled the number of themes (~50) you can apply. The new gestures make the app easier to use; the additional themes give you a lot of new geom-e-tree space to explore.

Minor changes and bug fixes:

Everything Else

More Help & Fun

On the internet...

Time Haven Media

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

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.

To Our Friends

Are you in a non-English-speaking country? Are you an artist seeking some kind of permission? A developer looking to imitate?

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!

Version 1.4, November 2013