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You are here: Home / Archives for applications

applications

AI Geometric Art Tools

Sunday, 24 August 2025 by Bruce Rawles

This post will pose more questions than answers, since it seems we’re still in the “wild west” early days of popular adoption of a plethora of AI offerings, and I am hardly keeping up with all the developments in so many sectors as “everyone and their uncle” jumps on the proverbial AI bandwagon!

(I’m also in the midst of moving from Arizona back to Oregon, so this will be a quick commentary! I hope to have much more useful information, helpful resources, and examples in a future post!)

In prior posts, I’ve dabbled with some AI-generated geometric imagery:

  • Digital dabbling with Dall-E (early primitive experiments)
  • AI generated geometry – revisited a few months later (a little further along, but still nascent)

Here is an example from the more recent post:

Dall-E 3 AI-generated image using prompt "make an image of (a) sea of mirrored geodesic spheres" thanks to Dave Van Dyke

Here is an example from the earlier post:

While the geometric imagery generated by artificial intelligence is often stunning, I typically notice that the images frequently don’t provide accurate models of precise geometric archetypes, being more oriented toward popular art and illustration than images for science, engineering, or renderings with CAD accuracy. If one looks closely, there are inconsistencies and “wonkiness” that don’t hold up to zoomed-in scrutiny if one wants the photorealism that ray-tracing software can achieve.

Do you know of websites, software, AI apps running on various platforms, browsers, devices, etc., that address this seeming void in AI functionality? I’m hoping that soon we’ll find mechanisms that assist in generating imagery with the same accuracy as ray-tracing software using precise geometric coordinates for the vertices of polyhedra, for example. If so, let me know via our contact form.

I found this web page of “AI Geometric Art Examples” by Open Art that looks promising, but when I click on the “… create …” buttons, I arrive at a more generic-looking interface. I’ll explore further when I have more time, but I’m eager to hear from readers if you’ve found good AI for geometry. Here’s a nice image from the page noted above:

AI Geometric Art Example by OpenArt
AI Geometric Art Example by OpenArt

I posed this query: “geometric image generation for science, engineering, renderings with CAD precision” to my browser (Brave) and got this response:

“Geometric Image Generation for Science
Recent advancements in geometric image generation for science, engineering, and high-precision renderings are increasingly focused on bridging the gap between visual data and parametric Computer-Aided Design (CAD) models. A significant development is the CADCrafter framework, which generates parametric CAD models directly from unconstrained real-world images, such as those captured by users with standard cameras.
This approach leverages a latent diffusion model conditioned on geometric features like depth and normal maps, which are invariant to texture and help mitigate the domain gap between synthetic training data and real-world images.
To ensure the generated CAD command sequences are geometrically valid, the model employs direct preference optimization (DPO) fine-tuning using feedback from an automatic CAD compiler, which acts as a code checker to categorize outputs as valid or invalid.

This method addresses the limitations of traditional generative models that produce 3D shapes with rough surfaces and indistinct edges, which lack the precision required for manufacturing and simulation.
By training solely on synthetic textureless CAD data and testing on real-world images, CADCrafter demonstrates robustness and generalization to unseen objects, even when processing single-view images.
The framework’s ability to distill knowledge from multi-view to single-view inputs enhances its accuracy and practicality for everyday applications.

Further, geometric deep learning (GDL) techniques are being applied to analyze and generate CAD data directly from complex representations like Boundary Representation (B-Rep), which encodes high-level parametric details.
GDL methods, including Graph Neural Networks (GNNs) and Transformers, are used for tasks such as similarity analysis, retrieval, synthesis, and autocompletion of CAD models, aiming to automate repetitive design tasks and facilitate design reuse.
These methods can learn intricate geometric features from B-Rep data without converting it to formats like meshes or point clouds, thus avoiding computational expense and information loss.

For applications involving imaging data, such as microtomographic scans, fast, realistic simulations on GPUs are used to generate synthetic CT images that include realistic artefacts like beam hardening and photon noise.
These simulations are crucial for creating accurate CAD models from data affected by strong imaging artefacts, enabling precise geometric analysis of materials and structures.
The gVirtualXray (gVXR) framework exemplifies this, simulating X-ray projections and CT volumes in milliseconds on a GPU, which is orders of magnitude faster than Monte Carlo simulations, while maintaining high accuracy.
This speed enables the use of repetitive simulations for tasks like image registration and training deep-learning algorithms.

These integrated approaches—combining image-to-CAD generation, geometric deep learning, and high-fidelity simulation—represent a transformative shift towards creating precise, editable, and manufacturable digital twins from diverse data sources, significantly enhancing workflows in science, engineering, and design.

AI-generated answer. Please verify critical facts.”

(Note the now familiar disclaimer in the last line.)


Do you have suggestions for future GeometryCode.com blog posts? Please let us know; we just updated and simplified our ongoing survey here.

Filed Under: 2D Geometries, 3D Geometries, AI-generated geometric art, applications, polyhedra, sacred geometry apps, sacred geometry news

Encyclopedia of Polyhedra By George W. Hart … and other geometric gems

Sunday, 16 March 2025 by Bruce Rawles

Apples and Oranges by George W. Hart https://www.georgehart.com/Applorng.html
Apples and Oranges by George W. Hart
https://www.georgehart.com/Applorng.html

It’s time to revisit the work of other polyhedra enthusiasts. Of particular note (having crossed my digital path again recently) is George W. Hart who has shared some excellent resources along these lines (faces, vertices, etc.) for decades. Here’s one gold mine to explore: Virtual Polyhedra: The Encyclopedia of Polyhedra By George W. Hart. Here is part of the table of contents and highly recommended if you want to explore the genres and categories of polyhedra; lots of great imagery, rotatable 3D models (rotate them with your mouse), facts and details:

  • Platonic Solids (Regular Convex Polyhedra) Background List of models
  • Kepler-Poinsot Polyhedra (Regular NonConvex Polyhedra)Background List of models
  • Archimedean Polyhedra (Semi-Regular Convex Polyhedra) Background List of models
  • Prisms and Anti-Prisms Background List of models
  • Archimedean Duals Background List of models
  • Quasi-Regular Polyhedra Background List of Models
  • Johnson Solids (the remaining convex polyhedra with regular faces) Background List of models
  • Pyramids, Dipyramids, and Trapezohedra Background List of models
  • Compound Polyhedra — Introduction Background List of models
  • Stellated Polyhedra — Introduction Background List of models
  • Compounds of Cubes Background List of models
  • Convex Deltahedra Background List of models
  • Zonohedra Background List of models
  • Uniform Polyhedra Background List of models
  • Uniform Compounds of Uniform Polyhedra Background List of models
  • Stellations of the Icosahedron Background List of models
  • Stellations of the Rhombic Triacontahedron Background List of models

Perhaps the most well-known and ubiquitous polyhedral shapes have fold-up models (and related patterns) for the first 3 categories above in Sacred Geometry Design Sourcebook (SGDS pages 196-220) and also on this website for the 5 Platonic Solids and 13 Archimedean Solids as well as Illustrations of Platonic And Archimedean Solids, Math Tables for Platonic And Archimedean Solids (and much more about polyhedra on this website.

compound of five cubes by George W. Hart
compound of five cubes by George W. Hart

If you exhaust this generous resource (or somehow overlook a few gems) here are more of his contributions – the videos are particularly helpful if you want to replicate his constructions:

  • The Pavilion of Polyhedreality
  • George’s Instagram page with some excellent 3D geometric sculptures
  • Mathematical Impressions: The Golden Ratio (short video)
  • Hyperboloids (short video)
  • Little Zonohedral Library (short video; my favorite of these which also features our mutual friend Russell Towle who showed me his 3D zonohedral models (such as Rhombic Spirallohedra)
    years ago in Dutch Flat, California)
  • Ceci n’est pas une lampe (fun, clever, short video)
  • Birdland (another clever short video)
  • Seven Slide-Together Constructions (another interesting short video)
  • … and MANY more!
Stained Glass Ball By George W. Hart https://www.georgehart.com/stained_glass.html
Stained Glass Ball By George W. Hart
https://www.georgehart.com/stained_glass.html

Filed Under: 2D Geometries, 3D Geometries, applications, Archimedean Solids, golden ratio, Platonic Solids, polyhedra, sacred geometry art

GeometryCode.com Survey

Friday, 1 December 2023 by Bruce Rawles

Thanks to all of you for your continued and ongoing support of this labor of love website!

It has been quite a while since we’ve had a reader survey or poll! Here are the results of the last one from the turn of the millennium:

 

Here’s an opportunity to share your geometric interests, passions, curiosities, favorite topics, and whatever you think will be helpful, useful, and fun!

Please share your interests, wish lists, and thoughts on existing and unexplored subjects via this survey to help guide our content priorities.

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Filed Under: 2D Geometries, 3D Geometries, 4D Geometries, applications, Archimedean Solids, audio, coloring books for adults, Fibonacci Numbers, Fractal Geometry, golden ratio, Hermetic Laws, modern physics, Numbers and Proportions, Platonic Solids, polyhedra, sacred geometry animation, sacred geometry apps, sacred geometry architecture, sacred geometry art, sacred geometry books, sacred geometry calendars, sacred geometry coloring books, sacred geometry interviews, sacred geometry jewelry, sacred geometry physics, sacred geometry toys, sacred geometry videos

Digital dabbling with Dall-E

Wednesday, 29 March 2023 by Bruce Rawles

After several years of mixed results with Siri (on my iPhone and MacBook Pro) and cylindrical kitchen add-on Alexa, often amusing and sometimes rather useful, I think my current favorite “Jetson” technology still might be our humble shorter cybernetic cylinder, “Rosie” the Roomba – our household vacuuming robot weekly whisking an amazing amount of fur from feline friends in our home.

Out of curiosity, I recently have done some very preliminary explorations with 2 OpenAI artificial intelligence apps; ChatGPT – rather impressive for certain well-defined tasks – and Dall-E 2. Since the latter seems appropriate for the visual nature of geometric imagery, I gave it a few prompts and got the results below within a span of about 45 minutes. Most of that time was spent thinking up a geometric ideas and then reviewing (clicking on the smaller thumbnails to see the zoomed-in) results; the actual time that the application took to generate the images with about 10-20 seconds for each batch of 4 rendered thumbnails! At its present maturity, Dall-E seems to have some limited usefulness if mathematical precision is less important than artistic conceptual prototypes. What do you think? Fun to play with, of course! The text I typed is followed by the output for each experiment:


experiment 1:
golden glowing dodecahedron hovering above a shimmering sea of 144 smaller iridescent silvery icosahedra


experiment 2:
a fleet of 12 toroidal UFOs arranged in a circle beaming light to the apex of the Great Pyramid of Giza, Egypt with a rainbow gradient sky


experiment 3:
Mandelbrot set made of cedar sprigs in the style of M.C. Escher


experiment 4:
(Anthony James mirrored icosahedron variations by Dall-E; I used this image as input)


experiment 5:
Impressionist painting in the style of Vincent Van Gogh of CERN hardware


experiment 6:
A photo of the Eiffel Tower scaled to fit in the middle of Stonehenge with glowing orbs atop each megalith in the style of Salvador Dali

… with 4 more variations requested:


experiment 7:
7 purple Flower of Life disks wrapped around a silver cone reflected in a parabolic mirror by Rene Magritte (and 2 more sets of 4 variations)


experiment 8:
a photorealistic image of Dorothy from the Wizard of Oz holding hands with Alice in Wonderland standing in front of the Great Pyramid at Giza, Egypt.
(If you look closely at the faces, they look rather wonky at best, but zoomed out not too bad… Of course the 2nd, 3rd and 4th images used the 2nd “Kephren” pyramid which still has the capstone casing and not the Great Pyramid, but this is a learning technology, right. I wonder if the next time someone asks for a similar image, Dall-E will incorporate my comments and stick to just the Great Pyramid?)


 

Filed Under: 2D Geometries, 3D Geometries, applications, sacred geometry apps, sacred geometry art, sacred geometry news

Sacred Geometry for healing

Sunday, 1 February 2009 by Bruce Rawles

Sometimes I get questions via email that seem appropriate for generalizing and sharing… Here’s one in that category. Thanks to all of you for over a decade for sharing your insights and musings!

Bruce,
Do you or anyone you know, have books on sacred geometry to heal physical and emotional issues…. any sacred geometrical patterns for detoxification????
I’d love to know!

Great question, thanks! (I’m taking the liberty of making this an anonymous blog post…)

I’m leaning more and more toward the idea that – here comes the ‘heresy’ for
geometry fans!  the particular form isn’t nearly as important as the feeling and intention of the person…

Having said that, I’d recommend using patterns and forms that have the greatest
resonance/meaning/kind & benevolent associations as possible… Aside from that,
I’ll keep my ears and eyes open… My interest lies mostly in the area of Universal
principles rather than emphasizing physical effects… It seems to me that if we
restore our awareness to wholeness (no separation), then whatever healing
can occur in the material world will be optimized.

I hope that’s helpful!

Cheers and blessings!

Filed Under: applications Tagged With: feeling, healing, intention, sacred geometry, wholeness

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