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3D Printing a Pendant with a Logo: A Mathematica-Based Workflow

A technical paper detailing a method to create 3D-printed pendants with custom logos using Mathematica, STL file generation, and multi-material printing.
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Home » Documentation » 3D Printing a Pendant with a Logo: A Mathematica-Based Workflow

Table of Contents

1. Introduction

This paper describes a project to manufacture a 3D-print of a pendant that includes a logo. The methods involve processing the logo image through a Mathematica script to create an STL file, which is then used by a 3D printer. The reader is assumed to be familiar with the basics of 3D printing. The project demonstrates a practical workflow for integrating 2D graphics into 3D-printed objects.

2. The Fighting Pancreases of Zachary University

JDRF (formerly the Juvenile Diabetes Research Foundation) is a major charity dedicated to preventing, treating, and curing type-1 diabetes (T1D). T1D occurs when the immune system destroys insulin-producing beta cells in the pancreas. About 1.25 million Americans have T1D. The author's family raises money for JDRF through the "JDRF One Walk" fundraiser. The team name is "The Fighting Pancreases of Zachary University", and the logo was designed by John and Xavier Golden (see Figure 1). The pendant created in this project features this logo.

3. Overview of the Design of the Pendant

The pendant is constructed using Mathematica by combining three parts: a base layer with "ZUFP" letters, a simple middle layer, and a top layer that is a 3D rendering of the Fighting Pancreas logo. Each layer is bounded by the circle $x^2 + (y+10)^2 = 4900$ and includes a hole for a clip defined by $x^2 + (y+64)^2 \leq 49$. The positive y-axis points downward due to how Mathematica stores image data. After joining the layers using Mathematica's Show command, the result is exported as an STL file for 3D printing.

4. The Base Layer – Mathematica

The base layer is defined for $0 \leq z \leq 6$. The creation begins by loading a JPEG file of the flipped "ZUFP" letters into Mathematica using code by Melissa Sherman-Bennett and Sylvanna Krawczyk. The code imports the image, converts it to grayscale, and processes the data matrix. The key commands are:

ClearAll["Global‘*"]
DataBase = Import["C:\\data\\3d\\ZUFP\\ZUFP-letters-flip.jpg"];
SizeBase = Import["C:\\data\\3d\\ZUFP\\ZUFP-letters-flip.jpg", "ImageSize"];
GrayBase = ColorConvert[Image[DataBase, "Real"], "Grayscale"];

The grayscale values (0 to 1) are used to determine the height of the base layer at each pixel.

5. The Middle Layer

The middle layer is a simple cylindrical spacer that provides structural support between the base and top layers. Its dimensions are defined by the same bounding circle and hole constraints. The height of this layer is typically set to a few millimeters to create a visible separation between the base text and the top logo.

6. The Top Layer – 3D Rendering of the Logo

The top layer is created by processing the Fighting Pancreas logo image through a similar Mathematica script. The grayscale values of the logo are mapped to heights, creating a 3D relief. The same bounding circle and hole constraints apply. The logo is oriented so that it appears correctly when the pendant is viewed from the front.

7. Combining Layers and Exporting STL

The three layers are combined using Mathematica's Show command. The combined 3D model is then exported as an STL file using the Export command. The STL file is typically large (over 20 MB) and should be saved locally to avoid network slowdowns. The model is scaled in the printer software to a diameter of 50 mm for a pendant, or smaller for earrings or charms.

8. 3D Printing Process and Material Considerations

The pendant is printed using three different colored filaments. The author uses a single-filament printer (Makerbot Replicator 2) and pauses the print at appropriate times to change filaments. Hatchbox filament from Amazon is recommended. The printing process requires careful timing to ensure proper layer adhesion and color transitions.

9. Technical Details and Mathematical Formulation

The bounding circle equation is $x^2 + (y+10)^2 = 4900$, which gives a radius of 70 units. The hole for the clip is defined by $x^2 + (y+64)^2 \leq 49$, with a radius of 7 units. The grayscale values from the JPEG images are linearly mapped to heights in the z-direction. For the base layer, the height $h(x,y)$ is given by $h(x,y) = 6 \times (1 - g(x,y))$, where $g(x,y)$ is the grayscale value (0 for black, 1 for white). This creates raised letters where the image is dark.

10. Experimental Results and Visual Documentation

Figure 1 in the original paper shows the Fighting Pancreas character, the original logo image, and the 3D-printed pendant from front and back views. The pendant successfully reproduces the logo with clear detail. The multi-color printing process results in a visually appealing object with distinct layers. The author reports that scaling the model to 50 mm diameter works well for a pendant, and smaller scales are suitable for earrings or bag tags.

11. Analytical Framework: Case Study of the Fighting Pancreas Logo

This case study demonstrates the workflow for converting a 2D logo into a 3D-printed pendant. The framework involves: (1) image acquisition and preprocessing, (2) grayscale conversion and height mapping, (3) layer definition and combination, (4) STL export, and (5) 3D printing with multi-material capabilities. The same framework can be applied to any logo or image, making it a versatile tool for personalized 3D printing projects.

12. Future Applications and Directions

The methods described in this paper can be extended to other applications, such as creating custom jewelry, promotional items, or educational models. Future work could explore automated color mapping, integration with dual-extrusion printers, and optimization of STL file size. The use of Mathematica for 3D printing opens up possibilities for complex mathematical shapes and data-driven designs.

13. Original Analysis and Expert Commentary

Core Insight: This paper presents a pragmatic, reproducible workflow for converting 2D logos into 3D-printed pendants using Mathematica, bridging the gap between image processing and additive manufacturing. The key innovation is the use of grayscale height mapping to create relief structures, which is a simple yet effective technique for personalizing 3D prints.

Logical Flow: The paper follows a linear, step-by-step structure: from problem motivation (JDRF fundraising) to technical implementation (Mathematica code, layer design) to practical execution (3D printing, filament changes). Each section builds logically on the previous, making it accessible to readers with basic 3D printing knowledge.

Strengths & Flaws: The strength of this work lies in its clarity and reproducibility. The code snippets and geometric constraints are well-documented, allowing others to replicate the process. However, the paper lacks quantitative evaluation of print quality, such as dimensional accuracy or surface finish. It also does not compare the Mathematica-based approach to alternative methods (e.g., using Blender or OpenSCAD). The reliance on a single-filament printer with manual filament changes is a limitation for scalability.

Actionable Insights: For practitioners, this workflow is immediately applicable for creating custom pendants, keychains, or badges. To improve efficiency, consider using dual-extrusion printers or automated filament changers. For researchers, the grayscale height mapping technique could be extended to incorporate color information or non-linear height functions. The paper also highlights the potential of Mathematica as a tool for 3D printing, which is underexplored compared to CAD software. Future work should focus on automating the multi-color printing process and evaluating the mechanical properties of the resulting objects.

14. References