We have added the external icosahedron tubular structure.
Though the tubes are all vertical and dont have the right size.
There are too much because we have not selected only the short edges.
The grids are displayed as 3 square grids at the equators.

import * as THREE from 'https://unpkg.com/three@latest/build/three.module.js?module';
import { OrbitControls } from 'https://unpkg.com/three@latest/examples/jsm/controls/OrbitControls.js?module';

// Golden Ratio and Scaling function
const PHI = (1 + Math.sqrt(5)) / 2;
const sqrt5 = Math.sqrt(5);

function transformCoord(k) {
    return (Math.round(k / PHI) + k * PHI) / sqrt5;
}

// Generate Icosahedron Vertices
const baseVertices = [
    [0, 5, 3], [0, -5, 3], [0, -5, -3], [0, 5, -3]
];

const icoVertices = [];
baseVertices.forEach(v => {
    icoVertices.push(v);
    icoVertices.push([v[1], v[2], v[0]]);
    icoVertices.push([v[2], v[0], v[1]]);
});

// Transform coordinates
const vertices = icoVertices.map(v => v.map(transformCoord));

// Parameters
let tubeThickness = 0.1;
let verticalTranslation = 0;
let gridOpacity = 0.5;

// Initialize Three.js Scene
const scene = new THREE.Scene();
const camera = new THREE.PerspectiveCamera(75, window.innerWidth / window.innerHeight, 0.1, 1000);
camera.position.z = 10;

const renderer = new THREE.WebGLRenderer();
renderer.setSize(window.innerWidth, window.innerHeight);
document.body.appendChild(renderer.domElement);

// Create Icosahedron Edges as Tubes
const material = new THREE.MeshBasicMaterial({ color: 0x00ff00 });
const tubeGeometry = new THREE.CylinderGeometry(tubeThickness, tubeThickness, 1, 8);

for (let i = 0; i < vertices.length; i++) {
    for (let j = i + 1; j < vertices.length; j++) {
        const midpoint = vertices[i].map((v, idx) => (v + vertices[j][idx]) / 2);
        const tube = new THREE.Mesh(tubeGeometry, material);
        tube.position.set(midpoint[0], midpoint[1] + verticalTranslation, midpoint[2]);
        scene.add(tube);
    }
}

// Create 3D Grid
const gridMaterial = new THREE.LineBasicMaterial({ color: 0xaaaaaa, transparent: true, opacity: gridOpacity });
const gridLines = [];

for (let k = -5; k <= 5; k++) {
    for (let i = 0; i < 3; i++) {
        let lineGeometry = new THREE.BufferGeometry();
        let points = [];
        for (let j = -5; j <= 5; j++) {
            let point = [0, 0, 0];
            point[i] = transformCoord(k);
            point[(i + 1) % 3] = transformCoord(j);
            points.push(new THREE.Vector3(...point));
        }
        lineGeometry.setFromPoints(points);
        gridLines.push(new THREE.Line(lineGeometry, gridMaterial));
    }
}

gridLines.forEach(line => scene.add(line));

// Controls
const controls = new OrbitControls(camera, renderer.domElement);

// Animation loop
function animate() {
    requestAnimationFrame(animate);
    renderer.render(scene, camera);
}
animate();