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Understanding the GeoCube
Shape Number
Shape Number:
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All together there exist more than 64 Possible Shapes.
The transformations follow a specific sequence starting with the Cube. The possible field of transformation is displayed to the right. The Cube is at the top of the image.
Each line stands for one single, direct transformation.
Since the transformations are non linear (in no specific order) there is no direct relationship between the shape number and its position in the Transformation Tree. Its sole purpose is for differentiation.
Possible Combinations
Possible Combinations:
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This number shows in how many ways the specific shape can "stick to itself". More specifically, how many different combinations exist, where the magnets will pull 2 identical faces together.
For example: The Cube has only one possible way to combine with another Cube. Square on Square. Whereas the UFO can be assembled in 2 ways. Inside on inside or outside on outside. Some shapes have more than 6 different ways of "self-attraction"
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This category is further divided into 2 categories: Simple and Complex.
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Simple: The 2 hemispheres of the GeoCube are magnetically identical. Less combinations are possible but each one will lead to a structure of higher complexity. Resulting in a 2,3 or 4 fold symmetry.
Each Shape can have between one and 7 possible combinations within the simple system.
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Complex: The 2 hemispheres of the GeoCube are magnetically polar opposites. More combinations are possible, most importantly towers and chains. Nonetheless more "dead-ends" exist, and the amount of possible arrangements can be overwhelming.
In this system, each shape can have up to 21 ways of connecting to itself.
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Hinge Sequence
Hinge Sequence:
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The hinge sequence is my approach of categorizing the different possible Shapes within the GeoCube.
In order to differentiate the shapes among each other one has to look at the different angles of the hinges that connect the pyramids.
Each GeoCube has 12 hinges. And each hinge can go through a motion between 0 and 270 degree. 0 meaning that the hinge is fully closed and 270 describing a fully opened hinge (as can be seen in the pictures).
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The sequencing of the hinges can be seen in the image below. Starting with a middle hinge the numbering alternates between top middle and bottom hinges.
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Middle; Top; Middle; Bottom; Middle; Top; etc.
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Each shape has a specific sequence of hinging, and following that sequence one can deduct and replicate each individual shape
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Most hinge positions are multiples of 90. Therefore I decided to further simplify the system by dividing the number in degrees by 90. This way each position can be described in a single digit term and for most of the shapes the denominators are either 0,1,2 or 3.
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Examples for the 5 Basic Shapes:
0 Degree
= 0
90 Degree
= 1
180 Degree
= 2
270 Degree
= 3
0 0 0 0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
2 2 2 2 2 2
2 2 2
0 0 0
0 0 0 0 0 0
2 2 2
2 2 2
2 2 2 2 2 2
0 0 0
2 2 2
2 2 2 2 2 2
Direct Relatives
Direct Relatives:
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In this field one can find other shapes that are only one transformation away.
Depending on the shape between 1 and 10 direct relatives exist.
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Multiple Cube Arrangements
Multiple Cube arrangements:
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In this field one can find Geometric assemblies that are created by 2 or more shapes of the same kind.
The small number in the corner indicates how many shapes are used in the creation.
Please keep in mind that this is not a complete list, but a display of some of my favourite combinations.
This gallery is intended to give and idea of what can be done with identical shapes alone.
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